in this video we're going to talk about how to find the inverse of a function so consider the function f of x is equal to two x minus seven what do we need to do the first thing that you should do is replace f of x with y y and f of x basically are the same thing now in your next step switch x with y so x is equal to two y minus seven and then after this step all you need to do is isolate the y variable solve for it get it by itself on one side of the equation so to do that let's add 7 to both sides so we're going to have x plus 7 is equal to 2y and to isolate y now we need to divide both sides by 2. so x plus 7 divided by 2 is equal to y so we can write the final answer as the inverse function is equal to x plus 7 divided by 2. and so that's a simple way in which you could find the inverse of a function but now let's look at some more examples try this one so let's say that f of x is equal to x cubed plus eight go ahead and find the inverse function so once again the first step is to replace f of x with y now the next step is to switch x with y so x is equal to y to the third plus eight finally solve isolate the variable y solve for it so let's subtract both sides by eight so we're gonna have x minus eight is equal to y to the third so how can we solve for y in this example what should we do next we need to get rid of this three we need to turn into a one so what we can do at this point is take the cube root of both sides so on the left we have the cube root of x minus eight on the right the cube root of y to the third is basically the threes will cancel it's three divided by three you get one so it becomes just y so therefore the inverse function is the cube root of x minus eight and so that's the answer now let's work on another example find the inverse function of the square root of x plus two minus five now go ahead and pause the video try this problem so let's start with the same process let's replace f of x with y next switch x with y so the steps are going to be the same so we're going to have x is equal to the square root of y plus 2 minus 5. next solve for y try to get it by itself so let's add 5 to both sides so we're going to have x plus 5 is equal to the square root of y plus 2. now we need to get rid of the square root on the right side so how can we do that how can we remove that radical so what we need to do at this point is we need to take the square of both sides of the equation so on the left side we have x plus 5 squared which is basically x plus 5 multiplied to itself twice so it's just x plus 5 times x plus 5. we just have two of them multiplied uh to each other on the right we simply have y plus two now on the left we need to foil x times x is x squared and then x times five that's five x and then we have another 5 times x and then it's 5 times 5 which is 25. so that's equal to y plus 2. next we need to combine like terms so let me just get rid of some stuff on top i'm always running out of space all right so here we go 5x plus 5x that's 10x so we got x squared plus 10x plus 25 and that's equal to y plus 2. now the last thing that we need to do is subtract both sides by 2. so we have x squared plus 10x and 25 minus 2 is 23 so that's equal to y so therefore the inverse function is x squared plus 10x plus 23. so that's the answer now going back to that same problem i want to show you something else now when we were at this step where we had x plus five squared is equal to y plus two if you choose not to foil this what you can do is simply subtract both sides by 2. so if we move this to the other side we're going to have x plus 5 squared minus 2 which is equal to y and so you could say that the inverse function is also equal to x plus 5 squared minus 2. so you could leave your answer like this if you want to but if you want to simplify it then you could expand this which we know it's going to be x squared plus 10x plus 25 and then minus 2 and say that the final answer is x squared plus 10x plus 23. so you can write the answer both ways because they're equivalent to each other so you have to pick and choose which way you prefer now let's look at another example this time we're going to deal with a cube root function so let's say that f of x is the cube root of x plus four minus two so go ahead and work on this problem so let's replace f of x with y as we've been doing before and then let's switch x with y so we have x is equal to the cube root of y plus four minus two and now let's solve for y so let's do that by adding 2 to both sides so we're going to have x plus 2 is equal to the cube root of y plus 4. now what do we do when we get to this part how can we get rid of the cube root symbol in order to get rid of it you need to take the cube of both sides so on the left side you're going to have x plus 2 raised to the third power and on the right side just y plus 4. these will cancel and now let's subtract both sides by four so we have x plus two raised to the third minus four which is equal to y and so we can write the final answer as the inverse function is equal to x plus two to the third power minus four and so this is the answer now if you want to expand it you could it might take some time but let's say if you're taking a multiple choice tests and you don't see this answer you may need to consider multiplying x plus two three times and then once you get a polynomial just subtract by 4 combine like terms and that's another way to express the answer as well but i'm not going to do that in this lesson now let's look at one final example so this one is going to be a little bit harder than the other ones so let's say that f of x is three x minus seven divided by four x plus three so like before we're going to switch f x with y and then we're going to switch x with y so everywhere you see an x replace it with a y now what do you think we need to do here how can we isolate the y variable what i would recommend doing is to write x as x over 1 and cross multiply so one times three y minus seven is three y minus seven and then we have x times 4y plus 3. so that's going to be 4y times x and then 3 times x now at this point what do you think we need to do in order to isolate the y variable what you should do at this point is you want to move every term that has a y variable on one side of the equation everything else that doesn't have that y variable move it to the other side so the 4y x i'm going to move it to the left side and the negative 7 i want to move it to the right side so i'm going to have 3y minus 4yx and that's equal to 3x plus 7. so as you move a turn from one side to the other side that term is going to switch sides so the 7 was negative on the left side but it's positive on the right side the 4yx was positive on the right side but it's negative on the left side so now that we have all the y variables on one side of the equation now we can take out the gcf we can factor out y so 3y divided by y is 3. negative 4y x divided by y is negative 4x on the right side we don't need to do anything at this point so now to get y by itself let's divide both sides by 3 minus 4x so y is equal to three x plus seven divided by three minus four x so here's the final answer the inverse function is three x plus seven divided by three minus four x and so that's the answer you