Domain and Range Functions & Graphs - Linear, Quadratic, Rational, Logarithmic & Square Ro

In this video we will focus on finding the

domain

and

range

of a function. We'll find it from a graph of

linear

quadratic

equations, equations that have

square

roots,

rational

functions

with fractions,

logarithmic

and exponential equations, so let's go ahead and To start, let's say that if we have a function that looks like this, how would you find the

domain

and

range

of this function so that this value is three? Here let's say this is approximately negative three and this is four and approximately negative 4. My graph is not perfect, so when you look for the domain you need to find all the possible values ​​of x that the function y has for the range that includes all the possible values ​​of y that the function contains, so what do you think is the domain for this function?

It's like this for the domain, you see it from left to right what is the lowest x value and what is the highest the lowest x value for the function is negative three and the highest is three, therefore the domain is minus three to three, so notice that i have parentheses on the negative three but I have a parenthesis on the positive 3. Whenever you have a closed circle you need to use a parentheses, but if you have an open circle make sure you use a parentheses, so now let's focus on the range that the range includes. all the possible y values ​​the function can have, so what is the lowest y value and what is the highest?

Notice that the lowest is negative four and the highest is four if we see it from the bottom up, therefore the range will be negative four to four, so in negative four we do not have an open circle, it includes negative 4 and the same occurs with 4, that's why we have a

square

bracket instead of parentheses, so let's try another example, what is the domain and range for this? Feel free to pause the video and try this example yourself, so let's start with the domain, what is the lowest possible x value that we have in this function?

Notice that we have an arrow instead of a closed circle. This arrow continues going to the left it goes to negative infinity and this arrow goes up but it also goes to the right, so x could be anything from negative infinity to infinity and x can even be two because at this point x is equal to two, then because x can be The real numbers will be from negative infinity to infinity. Now what about the range? So now let's focus on the y values. What is the lowest y value you see on this graph? Notice that the lowest y value is at negative three and the highest y value is. is positive infinity because it keeps going up forever, however notice that there is a break at two and continues at 4.

So between 2 and 4 there are no y values, so to write the range it will be the lowest y value which includes minus three. and then it goes up from negative three to two and it doesn't include two, so we'll use a parenthesis for that and then the union continues on four again and goes to infinity, but it includes four, so I need a bracket on four and it goes to infinity , so as you can see, it starts at minus three, stops at two, there's nothing between two and four, and then it starts again at four and goes up to infinity, and that's the shape you want. to see it when you're looking at the range, if you want to find the range, focus on the y values, if you want to find the domain, focus on the x values, let's try another example, okay, so for this function continue or for this graph. and go ahead and write down the domain and the range for this particular graph that you see, feel free to pause the video and see if you can get the answer, so let's start with the domain, so let's focus on the x values, so notice that we have an arrow. to the left, that means it will keep going down, but as it goes down, they slowly keep it to the left, so this graph can go left forever, so the lowest x value is negative infinity.

Now we don't really have a break at negative 3 because we see is between one and three, notice that there are no values ​​of x between one and three, the curve there is no red line between those two points, so we must eliminate that section of the domain and now we have an arrow that goes to the right, therefore the highest value of x is infinity and the lowest is negative infinity, so the domain will start from negative infinity all the way to one because x could be anything from negative infinity to one and then on. Union we have to start over at three because there is nothing between one. and three, so the union three to infinity, that is the domain of this function, so now let's focus on the range, the lowest y value is negative infinity because the graph keeps going down forever, although there is a break between these two curves that do not have a break between the values ​​of y in this curve and can be negative 2 and can be negative 2 in this curve, so y could be any value from negative infinity to 1. now between 3 and 1, if you notice that there is nothing between three and one, there is no red line, there is no curve between one and three in terms of y values, which means that for the range we must start from negative infinity and stop at one, then we must continue from one and stop at the highest point.

The value of y is approximately five, therefore the range will be negative infinity to one. Now include one because we have a closed circle and then we will need a square bracket since we have a closed circle and then the union now begins. We go back to three, but we have an open circle between three, so we'll need a parenthesis around three and then it stops at five. Now there's no open circle here, so it includes five and that's the range for this particular feature. Let's try another one. For example, let's say that if we have a vertical asymptote at x equals two and we also have a horizontal asymptote at y equals negative one and we have one graph that looks like this and another or another curve that looks like this but never touches the asymptotes, so with this type of graph, how would you write the domain and range?

So go ahead and take a minute and see if you can get the answer, so let's start with the domain so we can see that the function goes all the way to the left, therefore the lowest x value is negative infinity. Looking at it from left to right, we can see that the highest x value is positive infinity, but notice that the curve that never touches x equals two never touches the line, so we need to eliminate it. this x value of the domain, so the domain will be negative infinity up to two because it doesn't include two and then the union of 2 to infinity, so x could be anything from negative infinity to 2, but not itself and could be any higher value. 2 to infinity so now let's find the range for this graph so let's focus on the y values ​​let's look at it from the bottom up so the lowest y value is negative infinity because this graph keeps going down and the highest is infinite, however the curve never will. touches the horizontal line, therefore we need to remove the horizontal line or the horizontal asymptote from the range because y will never be negative one, so the range will be from negative infinity to negative one, union of negative one to infinity, so if you have horizontal and vertical asymptotes where the graph never touches those asymptotes for the domain remove the vertical asymptote and for the range make sure you remove the horizontal asymptote now it's not so bad to calculate the domain and range from a graph, but sometimes it's not You are given a Graph most of the time when you have an equation, so let's say if you have the equation y is equal to three x plus two, how can you calculate the domain and range from this graph?

So for

linear

equations it turns out that the domain and range are all real numbers x can be anything y y can be anything as long as you have a polynomial if the leading coefficient is one the range can be anything and for all polynomials the domain is negative from infinity to infinity, so let's graph the function, so this is a linear equation. in slope-intercept form y is equal to mx plus b, the slope is 3, the y-intercept is 2. so let's plot the y-intercept first at two and the slope is three, which means it's three over one, so which using lift over stroke we need to go up three units and one unit to the right, therefore the next point will be here and we can connect these two points with a line so that the graph looks like this, as you can see, it goes to infinity negative and to infinity without interruptions, that is why we can say that the range is from negative infinity to infinity.

Now, as it goes down, it also moves left and goes left forever, so the lowest x value is negative infinity. and the highest one is positive infinity because as it goes up it also moves to the right and that's why we can see that the domain is from negative infinity to infinity. Now, what about a

quadratic

function like y is equal to x squared plus two x minus eight? What is the domain and range of this function now? Whenever you have a polynomial, whether x raised to the first power x squared x cubed, if there are no fractions and if there are no radicals with an even index number or no absolute value, there are no records, there are no exponential equations if it is simply a polynomial the domain will always be from negative infinity to infinity because there are no restrictions on the value of x x anything, however if the leading coefficient is even then the range will be limited, the range will not be from negative infinity to infinity, but if the leading coefficient Usually it's strange that the range can be anything, so if it's like x raised to the first power which we covered in that example or x cubed, the range could be negative from infinity to infinity and you'll see that as we look at some examples, so let's move on. and graph this function, how would you graph y equals x squared plus two x plus eight?

So since x squared is positive, the principal function looks like this. One of the best ways to solve or graph it is to find the x-intercepts, if possible. Let's set it equal to zero and factor it: what two numbers are multiplied by negative eight but added to plus two this is going to be plus four and minus two four times minus two is minus eight but four plus minus two eh, that's what it adds up positive two, so to factor it will be x plus four times x minus two, so the x-intercepts are negative four and two, we need to set each factor equal to zero, so if you set x plus four equal to zero and if you subtract four from both the x sides are negative four and if you set x minus two equals zero and if you add two to both sides to solve x x is plus two then those are the x intercepts for this particular function, so if we plot the x intercepts we have one in two and another one at negative four now the vertex the x coordinate of the vertex will be the midpoint or the average of negative four and two the midpoint between negative four and two is negative one if you average those numbers if you add Separate them and divide them by two, you should get minus one minus four plus two is negative two minus 2 divided by 2 is negative 1.

So that's where the vertex is, that's the axis of symmetry, it's at x is equal to negative 1. Now you can also find the x coordinate of the vertex using this equation x is equal to negative b over 2a a is the number in front of x squared, which is one b is the number in front of times one, so negative two over two will be equal to negative one, which we have, so now you have two shapes and you have two shapes that you can use to calculate the x coordinate of the vertex. now once we have that, we need to find the value of y, so let's plug negative one into the equation, substitute x for negative one, so negative one squared plus two times negative one minus eight minus one squared, that's minus one times minus one, that's positive one. two times negative one, that's negative two minus eight, so we have negative one minus eight, which is negative nine.

Now it turns out that I need more points on this graph, so let's say that minus 9 is somewhere at the bottom, so at this point we can draw a rough sketch, so our graph will look like this, now we can calculate the domain and the range, so let's focus on the leftward domain, we have negative infinity, as the graph goes up it will go to the left. and then it can keep going to the left forever and all the way to the right we have positive infinity, there are no breaks in between, so we could say that x can be anything, the domain is from negative infinity to infinity, but now Let's focus on the range. because we have a quadratic function because it is a parabola, you can see that we have a minimum, the lowest y value is negative 9. is the x coordinate of the vertex if you have a positive x squared function if it is positive x squared you have a minimum value because it opens up, if it's negative x squared, you'll have a maximum value because it opens down, so the lowest y value is negative nine and then the function keeps going up forever, so it's positive infinity , so if you have a polynomial where the leading coefficient isEven typically your range has limitations, it won't be from negative infinity to infinity, so the range of the lowest y value is negative 9 and the highest y value is infinity and if infinity you should always put a parenthesis next to a bracket we never have a parenthesis for minus 9 because it includes minus 9. so this is the range and this is the domain.

Now let's say if you want to graph an equation in vertex form, let's say it's negative x plus 2 squared minus 1. How would you do it? graph it so that in vertex form we have the generic equation a x minus h squared plus k the vertex is h comma k so notice that h changes from we have negative h and h here so we need to reverse the sign here we have positive k and positive k, no we need to change the sign, so what is the vertex of this equation? So since we have an We know that it will open downwards, so at this point we can draw a rough sketch so that the graph looks like this, for any polynomial function the domain will always be from negative infinity to infinity, but now let's focus on the range, the value and lowest is negative infinity but the highest. is the y coordinate of the vertex is negative one, so the range is from minus infinity to minus one and that's all you have to do for this problem.

So what about this function y is equal to x minus two to the third plus one? domain and range for this function, so let's start with the parent function x cubed, the parent function looks like this, it is always increasing and this is positive x cubed, negative x cubed, it looks like this, the final behavior is towards up and then down, that's up and then down, so for both

functions

you can see that the domain is all real numbers x could be anything y for y the lowest y value is negative infinity and the highest is infinity , so the range is from negative infinity to infinity, so as long as you have a if the leading coefficient is odd, the range will always be negative from infinity to infinity for the most part and if you have a polynomial function, this will normally be the domain, so now let's graph the function with the transformation, because we see x minus 2 inside, if you set it equal to zero, you will get that x is equal to two, so it will shift two units to the right and we have a plus one on the outside, so it will increase by one unit, so the graph is will look like this, but as you can see, the domain is still negative from infinity to infinity, it can go all the way to the left and all the way to the right and, As you can see, the range the lowest y value is negative infinity and the highest is infinity, so if you got, say, some polynomial function x to the fifth plus four x cubed minus six x plus eight, you don't need to graph it if it is a polynomial function, the domain will always be negative infinity to infinity now for the range, the leading coefficient is x to the power of five, so if you look at the final behavior of the graph, when it is odd, if it is positive and odd, the final behavior will be like this towards the end, it will increase and to the left.

The side is decreasing, so we don't know what's going to happen in the middle, but you can see that the range will be negative from infinity to infinity if it's odd, so as long as you have a positive even function, the final behavior will be upwards. and up if it's negative it's down so the range has limitations for these you have to graph it to determine what the range is but if it's odd the range will be everything if it's negative and odd it will be from top to down, but the range will be negative from infinity to infinity.

If the leading coefficient has an odd exponent, what happens to the square root of x? What is the domain and range of this function? So if you were to graph the square root of x, it would look like this. Now you also need to understand the other transformations. negative square root x is reflected on the x axis and that's why it looks like this and negative square root x is reflected on the y axis so it looks like this and finally negative square root square of x is from zero to infinity but it includes zero the range the lowest y value is zero the highest y value is infinity so the range is also from zero to infinity but now if we focus on this function the domain is different the lowest x value is negative infinity but stops at zero, so the domain is negative infinity up to zero.

Now in the range we can see that the lowest y value is negative infinity, the highest is zero, so the range is also negative infinity up to zero. What about the negative square root x? the domain and the range for the domain, you can see that the lowest x value is zero, the highest is infinity, therefore the domain is from zero to infinity. now the range, the lowest y value is negative infinity, but the highest is zero, so the range is going to be negative infinity to zero now there's something else I want to mention with those four

graphs

, so let's start again with this y is equal to the positive square root positive x so notice that x is positive and y will associate it with the blue sign that is also positive when x and y are positive it will go to quadrant one because in quadrant one both this example x is negative but y is positive so x is negative to the left and is positive as you go up, so this graph will go towards quadrant two, where y axis, if x and y are both negative, then it will go towards quadrant 3. it looks like this and if y is negative and x is positive, when x is positive, you should go to the right, but if y is negative, you should go down, that's in quadrant four, so this graph will go this way, so let's say if you have the function y is equal to the square root of x minus two plus one and you want to find the domain and the range to find the domain of a function radical most of the time you can set the inside equal to or greater than zero then x will be equal to greater than two or you can graph it so you can see it shifts two units to the left and one unit up so it will start with two commas , one now x is positive and y is positive, so it will go towards quadrant one, so it will go this way.

Now this is enough to determine what the domain and range will be. We can see that the lowest x value is 2 and the highest is infinity, the lowest y value is 1 and the highest is infinity, therefore the domain is from 2 to infinity and includes two, the range is from one to infinity and includes one, so it might be easier to draw a sketch and then you can easily see what the domain and range will be. or you can set the interior equal to zero and solve for x, you can see that x is equal to or greater than two, which we can see that in a domain x is not less than two.

Now what about this function and is it equal to let's say? five minus radical x plus three, so how would you graph this function using the transformations? If you set the inside equal to or greater than zero, you'll see that 5 units, so it's going to start here at negative 3, point 5. Now, which direction is the graph going to go because that's going to be important to figure out the domain and the range, it's going to go towards quadrant one, towards quadrant two, towards quadrant three or into quadrant four, so we can see that we have a positive sign in front of x but a negative sign in front of the radical, so let's treat this as if we have a positive x value which means it will go to the right and a negative y value which means it's going to go down, so it's going to go towards quadrant 4 because x is positive and it's negative in quadrant 4.

So to draw a rough graph or a rough sketch, it will look like this. Our sketch doesn't have to be perfect, we just need to know where the first point is and what direction it goes, and that's all we need. Basically, find the domain and the range that we focus on in this video, so the lowest x value that you can see is negative 3 and it goes to infinity, so we can see that from this expression x is equal to or greater than negative 3. then, therefore, the domain includes negative 3 and goes to infinity. Now for the range, the lowest y value is minus infinity because as this curve goes to the right, it keeps decreasing and over time it's going to get smaller and smaller, so it's we're going to go down to negative infinity now the highest y value is basically this number, so the highest y value is five, so the range is from negative infinity and stops at five, but it includes five, so we need a parenthesis , let's try one more example and is equal to two minus the square root of four minus left or right? set the inside equal to or greater than zero and solve for x, so if we subtract four from both sides and negative x is greater than or equal to negative four and if we divide both sides by negative one, x will be less than or equal to four, so Note that whenever you multiply or divide by a negative number, the inequality sign must be reversed, so the graph will shift four units to the right, but the fact that x has to be less than or equal to four indicates It tells us that it will not go to the right but to the left, so we can go to quadrant two or to quadrant 3 because x has to be less than or equal to 4, so we won't get this. graph or this one now there is a plus 2 on the outside so it will shift up two units so the graph will start at four point two now let's determine the direction so that we have a negative sign in front of x and a negative sign in front of the radical, so negative x negative and x and y are negative towards quadrant three, so it will go in this general direction, so now we can calculate the domain and the range, so for the domain the lowest the highest is four, so it's from minus infinity to minus four, including minus four.

Now for the range, this function will continue to decrease as we move to the left, so the lowest y value is minus infinity, but the highest is two, so the range is from minus infinity to minus four. two and that's what you can do with that one so now let's go over some things with radicals so we can see that you can quickly find the domain by setting the interior equal to or greater than zero because you can't have a negative number inside a radical this doesn't is equal to a real number, is equal to an imaginary number 2i note that i is the square root of negative 1. so whenever you find the domain set the interior equal to greater than zero so here we can see that x is equal to greater than one, so the domain will be from one to infinity because you can only have positive numbers inside a radical if the index number is even, and if you don't see a number, now it's two to find a range, this number on the outside is important, if it's not there it will start or have a zero, but if it is there the range will include two, so since we have positive x and positive y, we know that the function is increasing. so therefore the lowest y value has to be 2 and the highest is infinity, so we could simply say that it is 2 from infinity.

Now sometimes you can get a graph that looks like this, so the highest value would be 2, the lowest would be negative infinity and for such a graph, the range will be negative infinity up to 2, but you still have both , so two should be somewhere in the range, that's why you need to know which direction the graph is going, whether it's increasing or decreasing. Sometimes you can have a radical. with an odd index number, let's say the cube root of x plus one, the cube root function of x looks like this, so since you have an odd index number, the domain could be anything because you can have the cube root of a odd number like real number is negative 2. cube root of positive 8 is positive 2. then you can take cube root of positive and negative numbers, square root of 25 is 5 but square root of negative 25 is imaginary number , is 5i, so that's Why you can't take the square root of negative numbers, it doesn't give a real solution and that's why if you have an even index number, the domain and range will be restricted, but if it's odd , the domain and range will all be real. numbers from negative infinity to infinity, so let's say that if we want to find the domain of the cube root of x plus one minus four to graph it, we're going to move one unit to the left and four units down, so the starting point should be somewhere around here, but it is It will have the same generic form, so it will look like this, as you can see, the domain is still from negative infinity to infinity and the range, the lowest y value is negative infinity and the most high is positive infinity, so every time it has a cube root. or the fifth fruit or the seventh root the domain and the rank will all beSo the graph should look like this based on the points so now we can find the domain and the range as you can see all we need to remove for the domain is the vertical asymptote minus three and we also need to remove everything I can't forget that one by one, so that's another thing, we have to be careful with this particular problem, so x is one somewhere around here, so I'm just going to put an open circle to Represent that we have a hole in one , so for the domain we need to eliminate negative three and one, so let me make some space for the domain to be from negative infinity to negative 3.

As you can see, the lowest x value is negative infinity and then the function will stop at negative three then start with negative three again, it will stop at one and then from one to infinity it will start again, so it's negative infinity to negative three union minus three to one from we have an open circle at one and then the union of one to infinity, so for the domain make sure to remove the vertical asymptote and any holes from the graph. Now what about the range we need to remove? the horizontal asymptote which is y is equal to two and we need to remove the y coordinate of the hole so we need to find out what the y coordinate of the hole is so we need to substitute 1 in the surviving equation which is the equation of 2x plus 1 over x plus 3, so it will be 2 times 1 plus 1 over 1 plus three, so two plus one is three, one plus three is four, so everything um it has the coordinates one point, three quarters, so the y value here should be approximately three over four, but keep in mind that I just drew a sketch, so we need to eliminate y is equal to two and three quarters of the range and keep in mind that the lowest y value is negative infinity and the highest is infinity, therefore the range is going to be from negative infinity to it will be three quarters or two, since three quarters is less than two, We'll get to three-quarters first, so it's negative infinity up to three-quarters, which is here and then after that.

It's going to be three quarters to two, so the union three quarters to two and after two it will go to infinity, so it's the union 2 to infinity, that's the problem with these bigger problems. I'm always running out of space. so, if you didn't understand that here is the range, it is negative infinity to three quarters union three quarters to two union two to infinity then we eliminate the y coordinate from the whole and also the horizontal asymptote from which we have to take it. the range, how can we find the domain and the range of the function y is equal to the negative absolute value of x minus three plus one?

So the first thing we need to do is know that we need to graph the absolute value of the main function. of x that is shaped like v looks like this, so now if we know the shape of that graph, we can graph the transformed function. If you set the inside equal to zero, you'll see that it will shift three units to the right and have a plus one on the outside, so the vertical shift is up, one so the vertex of the graph is three commas. Now notice that there is a negative sign in front of the absolute value, so instead of opening up, it will open up. down so it's going to look like this now with this information we have enough to graph it so as the function goes down it travels to the left and also to the right so x could be anywhere from negative infinity to positive infinity. for an absolute value function x to be anything the domain will be from negative infinity to infinity if there are no fractions and if there are no radicals now the range has limitations the lowest y value is negative infinity but the and higher is one, so the range is going to be from negative infinity to one and since it includes one, we need a parenthesis and that's how you can find the domain and range for an absolute value function.

Now what about an exponential function, let's say if you have the function 2 x plus 3 minus 1. now let's draw the main function and it is equal to 2 dx this graph looks like this this is the genetic form, but for most exponential functions you can simply connect points, the points I would choose will be negative 3 and negative 2 for x, if you are wondering how I got those points, set the exponent x plus 3 equal to 0 and 1. if you solve for x you will get negative 3 y 2 negative. Now if you plug in negative 3 for negative good number which is one now, if we substitute negative two for x, it will be two to the power of negative two plus three minus one minus two plus three is one, two raised to the first power is itself, which is two and two minus one is one , that's why I chose these values ​​because I'm going to get two good values ​​for y.

I can avoid fractions or large y values, so now it is much easier to graph the number on the outside of the negative that will be the horizontal asymptote that exponential functions have. Horizontal asymptotes and

logarithmic

functions have vertical asymptotes so the horizontal asymptote is and is equal to negative one so now let's graph it so first let's plot the horizontal asymptote that's where the function will start and we have the point minus three zero that is over here. and negative 2 1. so the graph will start from the asymptote and then increase, so what is the domain and range of this function?

The domain of an exponential function is all real numbers, it starts from the left and can continue going to the right forever as it goes to the right it will increase at a faster rate, so the domain of an exponential function is negative infinity to infinity. Now what about the range? The range of an exponential function has limitations and is limited by this number the horizontal asymptote, so it will start from the horizontal asymptote because that is the lowest y value and go to positive infinity because that is the highest y value as it increases, so the range is from negative 1 to infinity, but it doesn't include the horizontal asymptote, so now let's move on to a logarithmic function, let's say if we have log in base two x minus three plus four, what is the domain and range of this function?

A quick and easy method to calculate the domain is to set the inside greater than zero not equal to zero inside a record or a natural record cannot have any negative number and cannot have a zero inside it has to be greater than zero so just by looking at this the domain is x is greater than three, which is three to the power of infinity, but we'll show it with the graph now. The range of a logarithmic function is all real numbers. That's really all you need to do to calculate the domain and range of a logarithmic function that has the range. unrestricted the domain is restricted and just set the interior greater than zero and solve for x now to graph it what you do is set the interior equal to zero one and whatever the base is in this case two so this will give you the vertical asymptote because we said you can't have a zero inside the record because that's where the vertical asymptote will be, so the vertical asymptote goes is x is equal to three now the other two points will give us the values ​​we need for the table , then if x minus three equals one, if you add three to both sides, x will be four and if x minus three equals two, x is five, so now let's calculate the values ​​of y if we connect or if we replace x with four we are going to have log in base two four minus three plus four four minus three is one log in base one is always zero regardless of what the base is because two to the power of zero is one, so with zero plus four the first the point is in four now for the next one, let's connect well, let's replace x with five, so log base two five minus three five minus three is two, the log base two of two is one because two raised to the first power is two, so which is one plus four is five so now let's graph it so that we have a vertical asymptote at x equal to three and we have the point four point four which is here and five point five so for a logarithmic function the graph starts at the vertical asymptote and then continues to follow the two points, it will look like this, while for an exponential function it starts from a horizontal asymptote, as we can see, the lowest x value is 3, the highest is infinity, so the domain is 3 to infinity, as we said. before, x is actually greater than 3.

Now for range, the lowest value of y is negative infinity, but this keeps increasing forever, so for a logarithmic function y has no limitations, so range is from negative infinity to infinity, so now let's see if we can summarize. uh logarithmic and exponential functions so the generic 23x graph has a horizontal asymptote at y is equal to 0 and is an increasing function now the graph y is equal to log 2 of x will have a vertical asymptote at zero and is also an increasing function, so this is log and this is 2 dx, as you can see, an exponential function increases at an increasing rate, while a logarithmic function increases at a decreasing rate.

Now exponential functions have horizontal asymptotes, which means the range is restricted. You have to remove the horizontal asymptote if you want to write the range now for a logarithmic function, it has a vertical asymptote which means that the domain of a logarithmic function is restricted, so for exponential functions the domain is always negative from infinity to infinity , but for a logarithmic function. function the range is always negative from infinity to infinity now the range for an exponential function will start from zero and go to infinity since the horizontal asymptote is and is equal to zero now if it is different then this number will be different then if there is was a plus 3, the range would now be from 3 to infinity if increased.

Now note that if you have a negative sign in front of it, it will flip or mirror over the x-axis, so it will decrease, and in that case the range will be negative infinity to zero instead of zero to infinity. Now for the logarithmic function it has a vertical asymptote, so the domain is restricted, so the domain will be from 0 to infinity. Now, if you have a negative Now let's review trigonometric functions, what is the domain and range of sine x on the graph. of the sine it looks like this sine starts in the center, goes up and then goes down, the amplitude is one, so it varies between one and negative one now the sine wave goes on forever and goes both ways, so the domain of the Trigonometric functions for sine and cosine will always be negative from infinity to infinity, however the range has limitations, the range is based on the amplitude or y values, so the lowest y value is negative and the higher is one, so for the cosine the situation is the same. at the top it looks like this, but you can continue, but the amplitude is one, it will vary from one to negative one, the domain of a cosine wave will always be negative from infinity to infinity, that will be the same, the range in this case too is negative one to one, now sometimes it can have a vertical shift and the amplitude can be different, let's say if you have 3 or 3 negative sin x, let's make it positive 3 for now positive 3 sin x plus 2. then the graph shifts 2 units up, so the center will be at two, now the amplitude is three, so it will be three units above the center line, which has a maximum value of five, and three units below, so the sine wave will vary from minus one to five now the shape will be the same and continue forever, so as you can see the domain is still from minus infinity to infinity, but the range is now minus one to five and includes those points, so This is how you can find the values ​​for the range starting with this number: 2 plus 3 will give you the upper limit which is 5 and 2 minus 3 will give you the lower limit which is negative 1.

Let's try another example and it is equal to let's say negative 4 sine the upper limit we are going to add 4 to 3 which will give us 7 and to find the lower limit we are going to subtract 4 from 3 so that will give us a negative one now a positive sign that starts in the center then goes up, but the negative sign begins in the center and then down and as you can see the domain will be negative from infinity to infinity because the sine wave will keep going in both directions forever. The range dome is based on these. numbers, so the lowest y value is negative, one the highest is seven, this is how you can find the domain and range for a sine or cosine function.

Now the next thing we have to talk about is the tangent function. The tangent has a restricted domain, but the range is unlimited the tangent has a vertical asymptote at negative pi over two and at negative pi over two and at three pi over two and at negative three pi over two it has many vertical asymptotes the period is the distancebetween the two vertical asymptotes for a tan or cotan function the period is 2 pi over b where b is the number in front of x so b is 1 in this case well, actually let's go back to that for the sine and cosine waves the period is 2 pi over b but for a tangent wave or a wave cotan is pi over b so b is one so it is just pi which is the difference between pi over two and three pi over two now the tangent function is an increasing function it looks like this that is positive so negative tan looks like this as you can see it can go from negative infinity to infinity on the y axis, therefore the range is unlimited for the tangent or cotangent function;

However, the domain is everything except the vertical asymptotes, which will be difficult to write in interval notation, so, however you want to write it, I'll leave it up to you, but you have to remove the vertical asymptotes from the domain, so that It's all in this video. We've covered most of the topics you'll see regarding dominance and rank. We hope you found this video useful and thank you for watching and have a great day.