consider the function f of x let's say that it's equal to three x plus nine what is the inverse function given f of x how can we find it in order to find the inverse function the first thing we want to do is replace f of x with y so y is equal to three x plus nine next switch x and y so you need to understand what the inverse function is let's say if you have the point two comma seven the inverse point is 7 2. all you need to do is switch x and y so those points are inverses of each other so once you switch x with y or y with x go ahead and solve get y by itself so we need to subtract both sides by nine first and then to isolate y we need to divide by three so the function x minus 9 over 3 that's the inverse function that's how you can find it so now let's try another example let's say f of x is equal to x squared minus 4. go ahead and determine the inverse function feel free to pause the video so first let's replace f of x with y next let's switch x and y and then solve for the variable y so let's add 4 and then let's take the square root of both sides so the inverse function is simply the square root of x plus four that's the answer let's not forget to add plus or minus since we took the square root of both sides here's another one that you could try let's say that f of x is equal to the cube root of three x plus eight go ahead and determine the inverse function so let's replace y with f of x next let's switch x and y and then let's solve for the variable y so to solve it we need to raise both sides to the third power so x cubed is equal to three y plus eight the cube root and the cube will cancel so now we need to subtract both sides by 8 and then we need to divide both sides by 3. so now we have the inverse function so the inverse function is simply x cubed minus eight over three and so that's it consider the functions f of x which is equal to three x plus nine and g of x which is equal to x minus nine over three how can we determine if these two functions are inverses of each other here's what we need to do f of g of x we need to show that is equal to x and if we can also prove that g of f of x is equal to x then the two functions are inverses of each other so let's go ahead and do that so let's replace g of x with what it's equal to that's x minus nine divided by three and now let's take that and plug it into f so let's replace the x portion with what we see here that is x minus nine over three three divided by three cancels three divided by three is one so we have x minus nine plus nine negative nine plus nine is zero so we simply get x which is what we want now let's focus on the other one let's plug in f of x first let's replace f of x with three x plus nine and now let's plug this into g so let's replace x with three x plus nine so it's three x plus nine minus nine over three nine minus nine is zero so that leaves behind three x divided by three which is x so because both composite functions equal x f of x and g of x are inverses of each other consider the graph f of x is equal to x squared let's find the inverse of this function and let's graph it at the same time so y is equal to x squared to find the inverse we need to switch x and y and we need to find the value of y we need to isolate it so y is equal to plus or minus the square root of x which is the inverse function so the original function is f of x is equal to x squared and the inverse function is plus or minus root x this is the right side of the graph y is equal to x squared and the graph y is equal to the square root of x it looks like this let me draw that better notice that these two they're symmetric about the line y is equal to x that's a property of inverse functions the other side of y equals x squared is over here this is positive root x and then the other side of it negative root x is over here this portion is equidistant from the line y equals x they're still symmetric about it and so that's a property of inverse functions they're symmetric about that line so let's say if we have a function that looks like that and we want to draw the inverse function all we need to do is simply draw the reflection about that line so if the red line is f of x the blue line is the inverse function consider this function let me use a different color so let's say that's f of x the inverse of this function can you determine if the inverse is a function we know how to tell if f of x is a function if f x passes the vertical line test it's a function it touches it only once so f of x is a function now what about the inverse what can we do from this graph to determine if the inverse is a function there's something called the horizontal line test the graph touches the horizontal line test once which means that it's a one-to-one function for every x-value there's only one y-value and for every y-value there's only one x-value so if f of x passes the horizontal line test which means it's one to one that means that the inverse will pass the vertical line test which means the inverse is a function so let's say this is f of x is f of x a function does it pass the vertical line test and yes it does it only touches it once so f of x is a function now what about the inverse the inverse of f of x is that a function well let's see if f of x passes the horizontal line test it does not it touches the horizontal line more than once so because f of x is not a one-to-one function because it does not pass the horizontal line test the inverse is not a function the inverse will not pass the vertical line test so now let's prove it using a graph so let's draw the right side of y equals x squared and just the top part of y equals the square root of x that's y equals positive square root of x these two are inverses of each other but notice that this curve passes the horizontal line test it touched it once so the right side of y equals x squared is a one-to-one function and so the inverse which is y is equal to the square root of x notice that the inverse function it passes the vertical line test so because the original function passes the horizontal line test the inverse will pass the vertical line test which means that the inverse function is a function now let's draw the entire function y equals x squared not just the right side the inverse function of that entire graph is the blue line so notice that the function in red does not pass the horizontal line test it touches it at two points which means that it's no longer one to one and therefore the inverse function will no longer pass the vertical line test which means that the inverse function is not a function so you can use the horizontal line test to determine if the inverse function will pass the vertical line test if it passes it that means the inverse function is a function so if f x does not pass the horizontal line test the inverse function will not pass the vertical line test and so the inverse is not a function you