Finding Inverse Functions Algebraically (Step-by-step)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
in this episode you’ll learn how to find the inverse of a one-to-one function
we’ll go through the process step by step and you’ll see how easy it is
let’s do some math [Music] hi everyone welcome back um in this
episode we’re going to talk about finding the inverse of a function
now before we get started with that i wanted to mention that
in these videos here we’ve already talked about composition of functions
we’ve already talked about what inverse functions are and how to verify
and in the last episode we talked about what one-to-one functions are and we
looked at lots of examples so in this video we’re definitely going to
go beyond these three videos here these three episodes here so i recommend that
you check them out and give them a view the link is below in the description

00:01
so what we’re going to do is first you know ask the question what why is
the inverse function unique so we never really addressed that in these three
videos here and so i just wanted to kind of talk about
why the inverse function is unique so let’s say we have a function here
let’s call it f and let’s say we have inverse functions g and h are inverses of
f so f is given and we and and we’re uh saying g and h are inverses of f now um
because we often use the word the with the inverse function
so if i take two inverses of f how do we know that they’re equal to

00:02
each other and that it’s there’s only one inverse of it
so how do we show that two functions are equal so what we’re going to try to do
is to show that g of x is equal to h of x
and we’re not going to put any condition upon the x
and so this is going to be true for all x in the
domain that we’re going to be interested
in so i just want to kind of give you an idea of how something like that would
work now for this video right here we know
what it means to be an inverse function right here and from this one we know
what composition of functions is so to say that g is an inverse of f
what that means is that f composed of g this one is equal to x
and g composed of f is equal to x right here so these are
these two functions right here the composition are both the identity function
and we’re also assuming that h is also an inverse of f

00:03
so f composed of h of x is equal to x and also h composed of f of x
is equal to x so using this right here we can show that g of x is equal to
h of x but exactly how how would we do that so to do this we can
take this x in here and try to use something in here and try to get out an h so
um you know how would we do that so i’m going to look at this h right here
and say this x and i’m going to substitute in all this right here for it
and so let’s see what happens right there if i do that
so if i take this x right here and i just substitute all this in here
for the x so this will be f composed of h right here of x

00:04
now you might be asking why why did i choose that one why did i replace this x
with all of this i mean because we got the four things to use here to look at
um and so well i just want to pick this one right here because it’s going to
give us something useful so that’s a f with parentheses there yeah so this is
going to give us something useful something that we can look at here so
you’re welcome to try different combinations and in fact if you were to
start with h of x here you’d you’d probably do
use a different one of these but in any case um what happens if we replace this
x right here um right here with all of this right in here and then now
we’re going to get this is the function g composed of f composed of h of x
that’s just by definition um and um if we want to we can um [Music]

00:05
re use associativity so this will be g composed of f composed of h
and all that with an x here and now we’re going to use this one right here
which says this is the identity function right here so
this right here is going to be g composed of f composed of h of x right here
and so one more step here this right here is h of x
and the reason why is because g composed of f
if you give it any input you get out that
same input right there so if i give it h of x
it goes into this function right here you just get out h of x right there
so that uses these two right here um [Music] that just kind of gives you
an informal i don’t assume you have any background in writing mathematical
proofs at this point this is just a beginning precalculus series but this
kind of gives you an idea of why you would say the inverse function is unique

00:06
if you take two different inverse functions you can actually show that
they’re equal just by using the um definitions of what
it means to be inverse functions right there so if this gives you some kind of
idea right here of why this works right here um i use associativity and we
didn’t really prove that yet but in any case g is equal to h no matter
what the input is and so those functions are basically the same function right
there those are the same function all right so that kind of gives you an
idea of why we’re going to say the inverse function
is unique you know why we use the word v all right but anyways let’s get on to
some examples because uh in this video what we’re really concentrating on is
uh how can you find the inverse of a function so that’s the
that’s the topic of this video that was just a little aside there that that i
hope you enjoyed but anyways how can i find the inverse
of a function right that’s the question in this video here so uh let’s work out

00:07
some examples here so in the previous episodes we talked
about finding the inverse but we only did so informally so let’s look at a
more formal procedure now and so what i’m going to say is we have a function
right here and i’m going to give us a list of steps
that we can follow so the first step is to use x’s and y’s so i’m going to say
this is a y right here so that’s the first step is to set the function or
set it equal to y and you know anytime you have a function you can do that
so now what i’m going to do is i’m going to switch x’s and switch y’s wherever i
see a y i’m going to put an x and wherever i see an x i’m going to put a y
so this will be x equals and this will be 3y
minus 5. okay so that’s the second step first step set equal to y
second step switch x’s and y’s and now the third step will be to solve for y
so we’re going to solve this back for y again after switching now we’re going to

00:08
solve for y so to solve this for y i’m going to add the 5 over to both sides
i’m going to add the 5 over and then i’m going to simply divide by 3.
and this is the inverse function right here so i’m just going to name it so our
original function was f so i’m going to use f inverse right here and it’s just
going to be x plus 5 over 3 which is what you would expect the
inverse to be because if you’re going to give an input you’re going to multiply
it by 3 first and then you’re going to subtract 5. so we’re going to undo those
operations in the reverse order so here i’m going to add 5 first and then i’m
going to divide by 3. so these are inverses of each other
and we can verify that by checking these two
equations here checking this right here is the identity function
and by checking this right here is the identity function so we can check both
of those and we can verify and i’ll do something like that here in
a minute in case you missed that episode previously

00:09
so we could verify both of these and we would figure out that this this is the
inverse so this is what i haven’t talked
about so far in this series set it equal to y
switch x’s and y’s and then solve for y so this is the basic three steps right
there so in fact let’s just do one more quick one here let’s say f of x now
is equal to x to the third plus three all right so from the previous episode
we can graph this we can sketch the graph pretty quickly it’s just an x to
the third and then shift it up three then we can look at the graph and by the
previous episode we know what one to one function means
and we can see that it passes the horizontal line test so we can say hey
this function has an inverse function right so covered all that in the
previous episode check it out so yeah this is a one-to-one function it
must have an inverse and so now what’s new in this episode is how do you find
that inverse what is the step-by-step procedure well it’s the same every time

00:10
you set it equal to y we switch x’s and y’s so wherever you
see a y put an x wherever you see an x put a y and now we solve for
the y we solve for it and because we know this is one to one function
one two one one to one function the inverse has to exist so f inverse exists
passes horizontal line test all these three things mean the same
so when i set this equal to y that’s really just for uh ability to
manipulate algebraically right there’s really no
uh advantage to that other than just algebraic manipulation we switch x’s and
y’s that means we switched inverse relation
so now we have the inverse relationship but because we know all these three
things are true whichever way you want to think about
them we know the inverse must exist so that means we must be able to solve this

00:11
for y and we can in fact we can move the
3 over and then we can take cube root of both sides so when we move the 3 over
and then when we take cube root of both sides we have solved for y
and so now we have found our inverse function so this is the original
function with an f so i’ll use f inverse of x
and this will be cube root of x minus three and so there’s the inverse function
right there so again it’s just a simple matter of setting equal to y
switching the x’s and y’s and then solving for y now solving for y is
definitely the hardest step and so yeah let’s practice some more
so let’s look at um some more here and this time not only are we going to
find the inverse function but we’re also going to verify so we’re going to check
this out here so this will be a good review here alright so here we go

00:12
let’s look at this function let’s call it g of x is equal to let’s go with
cube root of x plus five and yeah let’s just go with that
function right there so cube root of x plus 5.
so let’s find the inverse function and then we’ll verify the inverse function
so to ver to find it we’re going to set it equal to y
and now we’re going to switch the x’s and y’s
and now i’m going to solve this for y now solving for y because this is cube
root so i’m going to raise both sides to the third power
and then i’m going to move the 5 over and so there is the new inverse function
right there so this was g is where we started so this one right
here is g inverse why is it g inverse because we switched
the x’s and y’s right there there’s where we did the inverse right there

00:13
all right so we set it equal to y switch x’s and y’s
solve for y and we have our new inverse function right there
okay so now let’s do the verifying part now that we found it now let’s verify so
i need to check these two things right here g composed of g inverse of x
equals x now we’re going to show some work
we have to end with x if we end with x we know we got it right we verified it
so let’s see what happens here so let’s use the definition of
composition so this is g of g inverse of x
and we know what g inverse of x is this inside part right here because we just
found it so this inside part right here g inverse
of x g inverse of x is this right here so this will be g of
g of x to the third minus five so i’m going to take x to the third
minus five and wrap that up and think about that as a whole thing that we’re

00:14
going to plug into g so what does g do it does cube root of so cube root of the
input and all of this is the input so x to the third minus five that’s the input
and then so it does the input plus five so the input plus five there we go so g
is the cube root of the input plus five so now let’s simplify this so this will
be cube root of and this is x to the third minus five plus five so just cube
root and then now that you see that this is just x
all right so we verified one of them so now let’s verify that
g inverse composed of g also works so this will be g inverse of g of x
so this will be g inverse of g of x we know is the cube root of x plus five
and so this now needs to go all of this needs

00:15
to go into g inverse so g inverse is going to cube something so it’s going to
be the input all of it cubed and then minus 5.
so these undo each other cube root and cubed and we get x plus five
and then minus five and that’s just equal to x
so we verified it this right here is equal to x
and this right here is equal to x and so there we go
we verified we found the inverse function and we verified the inverse functions
all right so [Music] um let’s do some more but i want to do
some that maybe are a little bit more challenging
let’s get some space right here [Music] so let’s do some more here
so let’s look at these two functions here and see if we can get them

00:16
so i’m going to go to [Music] 8x minus 4 i’m going to put the function
right here and we might need more space so let me see what we can do about that
let’s see if we can just go to there right there all right so there’s the
function right there let’s find the inverse function now
one of the things you may want to do is say uh find the inverse function right
well how do you even know there is an inverse function
so if you could graph this and see if it passes the horizontal line test
then you would know it has an inverse function so being able to graph this
is a is a nice thing to know how to do so you but if you don’t know how to do
it you could you know use a graphing program or a computer
or calculate a graphing calculator but in any case you look at a graph of this
and you realize oh it passes the horizontal line test right so it is one

00:17
to one and so the inverse does exist don’t just think that just because you
make up a function that the inverse has to exist it has to pass horizontal line
test then i know that it exists so i know what the graph this looks like
it does pass horizontal line test so i’m going to go and do the next step which
is to find it so don’t think you can find it first you have to know that it
exists all right so i know that the f inverse exists so i’m going
to find it now so i’m going to set this equal to y
and now i’m going to switch the x’s and y’s wherever i see a y i’m going to put
an x and wherever i see an x i’m going to put a y
there we go and now we’re going to solve this for y
now to solve this for y we’re going to need to multiply through here so i’m
going to say x times 4y plus 5 and then this is 8y minus 4.

00:18
and so this will be 4xy 4xy plus 5x equals 8y minus 4. so
we don’t have all the y’s on one side yet right so let’s move the 8y over here
and let’s move the 5x over here so i’m going to say this is 4xy
and then minus 8y and then say this is minus 5x minus 4.
now out of these two we’re going to factor out the y so i get 4x minus 8
and minus 5x minus 4. and now the last step is i’m going to
take this right here and divide it underneath so we’re going to get
minus 5x minus 4 over four x minus eight and now there’s three negatives so
maybe we can write it more positively you know
factor out a minus one on top factor a minus one out on bottom and cancel the

00:19
minus ones and then we get five x plus four over um eight minus four x
so that would be a nice way to write it right there so we found the inverse
function right here we’ll go put it up here so f f inverse is 5x plus 4
and then 8 minus 4x there we go oops can’t quite see that so let’s go
with 5x plus 4 and then 8 minus 4x all right very good so here’s the function
and here’s its inverse and now let’s verify now technically you
don’t need to verify unless someone just asks you to do it but i’m going to
go ahead and verify this right here and see this right here
um yeah let me get rid of that there we go so let’s verify here so

00:20
i’m going to do f of f inverse of x and if this is right if this worked
this should get x it may take a lot of work
but we should get x so dot dot lots of work we should get equal to x when we’re
done let’s see if this works so this will be f of f inverse of x
maybe i can go one more over here uh so this will be f so what’s the
inside part of the inverse f inverse of x it’s all of this so this will be 5x
plus 4 over 8 minus 4x and then let’s change the size here there we go
so f of f inverse so f of f inverse and now i need to take all of this and
substitute it into f so let’s do that say about right here
so think of all of this as all rolled up and put into an input so what does f do
when you give it an input i can put it here and i can put it here

00:21
so i need to put all of this right here and right here so here we go so 8 times
all of this 5x plus 4 and then 8 minus 4x so 8 times x minus 4
and then 4 times the x the input which is all of this so 5x plus 4
all over 8 minus 4x so 4x plus 5. so 4x plus 5. there we go
so all this should be equal to x right so obviously it’s equal to x right
all right let’s see what we can do um notice here the denominator is 8 minus
4x the denominator is 8 minus 4x but these two don’t have a denominator of 8
minus 4x you can think of this as 4 over 1. so here’s what we’re going to do

00:22
8 times 5x plus 4 minus 4 times 8 minus 4x all of that over 8 minus 4x
all of that’s over and then here we have 4 times this part and then 5x plus 4
minus 5 plus 5 so here i’m thinking about this as 5 over 1
and i’m just multiplying by 8 minus 4x top and bottom so i have a common
denominator so this 5 is going to change into this so this will be 8 minus 4x
and then i have 8 minus 4x on the bottom and an 8 minus 4x on the bottom so i
just use common denominator here all right so here we go so these cancel now
and so let’s see what we’re going to get on top here we’re going to get a 40x
and then a plus 32 and then a minus 32 [Music] and then a 16x perfect

00:23
now here we’re going to get a 20x and then a plus 16 and then a plus 40.
and then a minus 20 x all right so let’s see what happens here
so these two add up to zero so on top i’m going to get a 56x
and in the denominator so these right here cancel
the 20x and the minus 20x they add up to zero so this gives us a 56
and now the 56 cancels and voila we get our x perfect
so we verified that f composed with f inverse that that’s the identity function
whatever you plug in x is what you get out x
all right so we verified one of the two we need to do
so now let’s go do the other one so let me erase this real quick all right good

00:24
so now we need to do the f inverse of f at x
and so now f is going to be applied to x first
so f of x goes first so now i’m going to use this one in here first so this will
be f inverse of all of this 8x minus 4 over 4x plus 5. very good
all right so now we’re going to take all of this right here wrap it all up as a
big o x and put it into f inverse so this is my input all of this
so what does f inverse do to an input it multiplies a five and adds four all
right so we’re gonna do five and then all of this eight x minus four
over four x plus five plus 4 all right we did the top here and now 8
minus 4 times the input 8 minus 4 times the input which is 8x minus 4

00:25
over 4x plus 5. and so as you see we’re going to do the
same thing that we did on the last one a second ago
because what we’re going to need is a common denominator of 4x plus 5 on this
one so i’m going to think of this 4 as a 4 over one and and then multiply
top and bottom by four x plus five and so what we get here is i’ll do this
on the next step here so we’re gonna get five times eight x minus four
plus four times four x plus 5 all over 4x plus 5 and then all this is over
8 times 8x minus four over four x plus five so it’ll be eight times all of that
and then so minus wait sorry uh what am i doing here okay okay that’s not right

00:26
so sorry about that so this is going to be 5 times that oh
that’s over the denominator okay yeah sorry sorry we’re getting common
denominator so yeah this will be 8 times 4x plus 5 and then minus 4 times
8x minus 4 and then all over 4x plus 5. all right so
now we can cancel these right here and you know actually this compound fraction
i’m going to rewrite it because whenever you divide you’re just multiplying by
the reciprocal which is 4x plus 5 over all this right here
which is 8 times 4x plus 5 minus 4 times eight x minus four
all right so now i can erase this here and have enough room

00:27
so you see the four x plus fives cancel let’s see what we’re going to get here
we’re gonna get a 40x minus 20 and then a plus 16x plus 20.
and so that canceled with that and so then on the denominator we’re going to
get 32x and then plus 40 and then minus 32x
and then a minus 4 times -4 so plus 16. all right so now let’s see what’s
happening here the 20s add up to zero and so we’re going to get 56 x
and down here the 32 x’s add up to zero and we get 56 so we get 56x over 56
again which is just x all right so this this composition right
here is also the identity function and so we did it we verified so we found

00:28
the inverse right here and then we verified using this and the other part
that we erased there so that’s good all right so now um let’s see here we
have this part right here which is this g of x right here um so i’m going to
we’re going to find the inverse first let’s find the inverse of this right here
so here’s uh g of x right here so i’m going to call this equal to y so 6x plus 4
2x plus 6 here now look at all those twos
i don’t know would it be easier to just work with smaller numbers 3x plus 2
cancel a 2 everywhere x plus 3 right so let’s just look at that function there
is this the same function right cancel a 2 everywhere in any case

00:29
i’m going to switch x’s and y’s now so three y plus two and then y plus three
there we go and now i’m going to solve this for y
so to do that i’m going to multiply y plus three
and then we get three y plus two multiply both sides by y plus 3 let’s
say and then xy plus 3x equals 3y plus 2 very good
so now i’m going to move the 3y over here so i’m going to say xy minus 3y
and minus 3x plus 2 so we’re going to set it equal to y
switch x’s and y’s and then solve for y we almost have it solved for y i’m going
to factor out the y x minus 3 minus 3 x plus 2
and then now i’m going to divide so we’re going to get y equals minus 3x plus 2
over x minus 3. and now there’s only two negatives and two positives so maybe

00:30
we’ll just leave it the way it is all right so there’s g
and so here’s g and so here’s g inverse now maybe you want to put the
twos back in everywhere and say minus 6x plus 4 and then 2x minus 6. but i mean
it doesn’t really matter in fact i’m just going to say over here that g of x
is equal to 3x plus 2 and then x plus 3 and g inverse is the minus 3x plus 2
and then the x minus 3. there we go and so let’s verify now
find the inverse and verify and so let’s verify
and this time i’ll try to pick up the pace a little bit
um so i’m going to do g of g inverse of x

00:31
and so this will be g of g inverse of x and so this will be g of so g inverse is
this one right here minus 3x plus 2 over x minus 3 [Music]
and so now i’m going to put all that into g so here’s g right here so 3
and then all of this minus 3x plus 2 over x minus 3 and then plus 2
and then all over x which is minus 3x plus 2 over x minus 3.
so there’s the x and then plus 3. okay so now let’s get a common denominator
in fact i’m going to get a common denominator and cancel
all at the same time so i’m going to say 3 times the minus 3x plus 2
and then plus two times x minus three all over so that’s um plus

00:32
two times x minus three and then all over okay so all over
so i’m gonna get a common denominator and we’ve already canceled the x minus
threes so i’m going to get a minus three x plus two right here for that part
right there plus three times the top and bottom x minus three so x minus
three right here right there we go so on top we’re getting minus nine x plus six
and then plus two x minus six and then minus three x plus two
and then times three x and then minus nine all right so the sixes add up to zero
and so we’re going to get minus seven x on top
and then here with the minus three x the
three x’s add up to zero and we’re going to get minus seven in the denominator
and voila we got the x now i’d like to just reiterate that it
says to find it and verify so finding it was
crucial because this is a one-to-one function

00:33
and you know this function right here you can look at the graph and see that
it passes horizontal line test so it is one to one and so we found the inverse
and this right here is the inverse and it says to verify so once we if it says
to verify then we can verify but we don’t really need to verify because the
process of finding the inverse always works once you know it it does exist
then it will always work so verifying it is just because they asked us to do it
so we did half of the verifying now let’s do the second half
so the second half is when g inverse gets applied to g of x
so this will be g of g inverse of g of x and now let’s substitute in here what
did g of x is so that’s the 3x plus 2 over x plus 3. [Music]

00:34
so now all of this goes into g inverse i’m going to get minus 3 times the input
so minus 3 times the input plus 2. all over the input -3 so the input right here
and then the minus 3 there so again i’m going to get the common
denominator of x plus three so we need to multiply top and bottom by x x plus
three and multiply top and bottom you know think of that as top and bottom
right so by x plus three so we’re gonna get minus nine x and then minus six
and then two times the x plus three so that’ll be plus two x and then plus six
and so now this will be three x plus two and then minus 3 times the
x plus 3 so that would be minus 3 times x
and then minus 3 times -3 so that would be minus 9.

00:35
and so these add up to zero and i get minus seven x
and these add up to zero and i get minus seven here and so in the end we get out
an x wow that’s nice all right so there we go i hope that helps you
able to find the inverse and we also you know practice the composition some
more and practice verifying and so i hope that you enjoyed this video
and i look forward to seeing you in the next one take it easy bye bye
if you enjoyed this video please like and subscribe to my channel
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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