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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
