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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 2x minus 7 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 2y

minus 7 and then after this step all I

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 gonna have

X plus 7 is equal to 2y and to isolate Y

and 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 8 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 8 finally solve isolate

the variable Y solve for it so let's

subtract both sides by 8 so we're gonna

have X minus 8 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 3 we need to turn

into a 1 so what we can do at this point

is take the cube root of both sides so

on the Left we have the key

of X minus eight on the right the cube

root of Y to the third is basically the

threes will cancel is 3 divided by 3 you

get 1 so it becomes just Y so therefore

the inverse function is the cube root of

x minus 8 and so that's the answer

now let's work on another example find

the inverse function of the square root

of x plus 2 minus 5 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 gonna 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 to each other on the right

we simply have y plus 2 now on the Left

we need to foil x times X is x squared

and then x times 5 that's 5x 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 are at this step where we had X plus

5 squared is equal to y plus 2 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 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 4

minus 2 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 swiftlife so we have X is

equal to the cube root of y plus 4 minus

2 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 4

so we have X plus 2 raised to the third

minus 4 which is equal to line and so we

can write the final answer as the

inverse function is equal to X plus 2 to

the third power minus 4 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 test and you don't see

this answer you may need to consider

multiplying X plus 2 three times and

then once you get a polynomial just

subtracted by four 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's going to

be a little bit harder than the other

ones so let's say that f of X is 3x

minus 7 divided by 4x plus 3 it's like

before we're going to switch f of 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 1 times 3y minus 7 is 3y

minus 7 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 we 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 4yx I'm going

to move it to the left side and the

negative seven I want to move it to the

right side so I'm gonna have 3y minus 4y

X and that's equal to three X plus seven

so as I move a term from one side to the

other side that term is going to switch

sides so the seven was negative on the

left side but it's positive on the right

side the 4y x 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 a Y

so three Y divided by Y is three

negative four Y X divided by Y is

negative four X 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 three minus four X so Y is

equal to 3x plus 7 divided by three

minus 4x

so is the final answer the inverse

function is 3x plus 7 divided by 3 minus

4x and so that's the answer