What Are Inverse Functions and How to Verify (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 what the inverse function is and how to verify
whether or not two functions are inverses of each other let’s do some math
[Music] hi everyone welcome back i’m dave
uh let’s start talking about what is an inverse relation
and so i’m going to do that by um kind of looking back on what we talked about
when we first talked about what a function is
so let’s recall what a relation is so for example
we can say this right here is a relation so i’ll put in some ordered pairs here
so here’s a set of ordered pairs it’s got three ordered pairs
and i’m naming the set s and we can talk about this being a relation

00:01
so is a relation simply because it’s a set of ordered pairs so um we have one
can be the input into s and 2 would be the output minus 1 would be the input 3
would be the output and 4 would be the input and 5 would be the output
so associated with the relation we have the inputs and we have the outputs
now to talk about inverse relation so here’s the notation we would use for
inverse relation and so what we’re going to do is we’re
going to switch the inputs and outputs so this ordered pair becomes the ordered
pair 2 1 and this ordered pair becomes 3 minus 1
and this ordered pair becomes 5 4. and so this is also a relation
now we use this notation right here not to mean like one over or like it’s

00:02
an exponent but it just means it’s the inverse that we switched the x’s and y’s
the inputs and outputs so these are inverses of each other
inverse relations of each other because if i do the inverse of this inverse i’ll
get back to the original right here so we can say s is a relation and s inverse
is the inverse relation of s so we could say s inverse is the inverse relation
of s and we can repeat that process we could say s inverse s inverse is the
inverse relation of f inverse and so this is just s
because i would just switch and then i would switch back
right so that’s equal to s so s is the inverse relation of s inverse and s
inverse is the inverse relation of s now nowhere in here am i talking about a

00:03
function though it’s just all about relations so this episode is about inverse
functions so let’s talk about that now so this is an inverse this is a relation
the question can be asked is it also a function
so for each input we have to have a unique output and that actually works
doesn’t it because if i input one i get only one output if i input minus one i
only get one output so if i added something like this to the relation here
so i’ll say here what if we added the pair minus one five
so this is still a relation is a relation but notice that this right here is
not a function right so this would be the point of switching
x’s and y’s this would be minus 1 1. so this is also a relation
i forgot to put parentheses there when i

00:04
closed about before but in any case this is s now and this is s inverse there’s
still relations and these sentences still hold
but now when i ask the question is this a function is s a function now i have to
say no because if i input minus one i get two different outputs i don’t get a
unique output i get three is an output and a five is an output
so this is not a relation uh this is not a function right here so not a function
now what happens if i take that point back out
so let’s take that back out now and now let’s ask the question again it’s still
relation but now let’s ask is this relation a function
now we can see this is a function and so we can ask the question
is this function does this function have an inverse function
and so now i’ll look at the inverse and look at the inverse relation right here

00:05
and i ask the question is the inverse also a function we know the original one
is a function but now i’m asking is the inverse a function so input two i only
output one thing i input three i only output one thing
i input five and i only output one thing so this is also a function
so not only is this right here an inverse relation of s
but it’s also an inverse function of s it’s the inverse function of s
now what happens if we put this point right here in here let’s say we have 3 5.
here we go so now is s a relation yes it’s a set of ordered pairs
now is this a relation a function and i’ll have to check four things i’ll
have to check four inputs remember you have to check every input so i check i
get only one i get only one output i get

00:06
only one output i get only one output it doesn’t matter that four and four and
three share the same output what matters
is does every input have only one output so this right here is a function and so
now i’m going to go is a function and so now i’m going to go and try to find the
inverse and ask if it’s a function or not so now i’ll have to put in here 5 3
to find the inverse so here’s s and here’s s inverse and so now i’ll ask the
question is it a relation of course yes is it a function
is the inverse a function and now we’ll have to say no for this example here if
we added this point in here 3 5 when you switch it around and when you switch
this one around now we can say 5 has a unique output no 5 has only
five has two different outputs so this is a relation but not a function
but not a function so here’s an example of a relation that is a function

00:07
so here’s an example of a function that does not have an inverse function
its inverse is just a relation so i hope that helps
but now we’re not going to just work with a finite number of points one two
three four points you know how can we work with infinitely many points how can
we work when we have function rules so um let me try to go with a
definition now of what an inverse function is now so let’s uh get this out
here real quick so what is an inverse function so we’re going to start with two
functions right here let f and g be functions now there should be an s right
there let f and g be functions and so now we’re going to say that
these two right here are inverse functions of each other and to do that

00:08
we’re going to define what that means there’s the yes if
so in the last episode we talked about composition so
i’ll write the composition notation out here that we talked about in the last
episode so this is the composition notation right here
so yeah before i go on let me just mention here that this episode is part
of the series functions and their graphs step-by-step tutorials for beginners
link below in the description for the full series but in the last episode we
talked about the composition of functions and this is the definition of
it right here so what does it have to do with inverses well if you input an x
and g does something to that x maybe it squares it or cubes it or subtracts
three whatever g does to that x then f will undo
what g does in other words whatever input you get in

00:09
you get that out also so if i input a 2 that 2 will go into g g will do
something to the 2 and then f will undo what g did and you get out back to 2.
so you have if that condition holds and this condition holds in other words
it doesn’t matter which way you go if x goes into g first then f will undo it or
if x goes into f first then g will undo what f does
so you have to get the identity function
out in both of these cases right here so and again in the previous episode we
talked about what the composition is we we worked out lots of examples with
this is the composition right here so if this composition right here is just the
identity function just you put in you get out you put in you get out and this
has to happen for every x in the domain of g
right because x is going into g and this right here has to happen for every x in
the domain of f all right so if those two conditions hold then what

00:10
we have f and g are called inverse functions of each other and we have
special notation for that which i’ll show you in a moment
so let’s just do a quick example right here so let’s say we have the domain of f
so let’s just call this domain of f and let’s say we have an x right here
and so let’s say over here we have the range of f range of f
and let’s say we have an output over here f of x and f is going to be
x plus four so what function do you think will undo x plus four
so we’re going to have a function right here called g
to undo the x plus 4 it’s going to do the x minus 4.
so these functions undo each other for example if i input a 7 i’ll do 7 plus 4

00:11
and g will say oh no you don’t take and take away that 4 and you get back to the

  1. so we can we can check that we can say f composed of g of x will be
    f of g of x which is so g of x is x minus four so i’ll put in here x minus four
    and now i take x plus four all into f so x minus four and then plus at four
    and so the fours undo each other i don’t need this here
    and so this would just be x and then we can check this one right
    here also g composed of x uh sorry g composed of f
    of x so this will be g last f first so what is f of x f goes first so it’s x
    plus 4 and now i put x plus 4 into g so x plus 4 and then minus four

00:12
so that’s the input and then minus four and the four is still undo each other
and they add up to zero so x plus zero in other words x so
both of these work right here and so yeah we just have the domain
and then we map and then we come back over here and so this right
here is the domain of f but you could also say i’ll make it a little bit
bigger you can also say that this is the domain of g and
so i’ll say domain of f and then i’ll say a little bit bigger here domain of f
and this will be the range of g and so here i’ll just put a little x here
so yeah the domain of f range of g range of f domain of g and so we can go
and and map this right here to an f of x

00:13
and then we can take that f of x and map it right back to that g
and that would be the composition right there one composition and then you have
the other composition also all right so yeah let’s look at another example here
here we go let’s let’s take a look at this right here
so which of the following functions is the inverse function of this right here
so what would you guess so would you guess a or b or the function g
what is that you think is the inverse or do you think h is the inverse
so think about how to undo this if someone put an x in here how would
you undo this so for example if i input a 3 what would i get here
or let’s say i input a 1. if i input a 1 here first thing i’m
going to do is subtract the 5 and then i’m going to divide
and then multiply by 2. so how would i undo those operations how

00:14
would i do them backwards and undo them so do you think you’re going to take an
x first subtract a 5 that doesn’t seem to be undoing this right here i think
i think um actually that i think this um should be a 5 right here actually
and so let’s make that a 5 right there so let’s see which one it is here let’s
let’s try g first here so i’m going to try g just to see i don’t think it’s g it
doesn’t seem very intuitive to me that’s g because to g here just looked like it
just flipped it up upside down right and that’s not really what
inverse means inverse means undo it it doesn’t mean flip it right that’s
completely two different things for example in the last episode we said
add four subtract four that’s an undo operation that’s not a flip operation

00:15
right so i don’t think it’s g or part a so let’s check though what
happens if we do f composed of g of x if we get out an x right if we do work
work work we get out an x i don’t know maybe it is g
let’s see if work work work will give us the x so here we go f of g of x
and what is the g of x here it’s x minus five over two
and now if i do x minus five all of that
into f i’m going to get 2 over the input which is x minus 5 over 2
and then minus 5. now is all that equal to an x
well it’s maybe it’s a little bit hard to say
maybe let’s you know simplify it and see
maybe maybe it’s an x who knows so let’s just say here this is
um 2 over x minus 5 over 2 and then let’s go with 10 over 2.

00:16
instead of a 5 here let’s say it’s 10 over 2. now i have a common denominator
here so 2 over so this will be x minus 5 minus 10 and then all over two
and then so this is two over or let’s say two times two over one
times and then now let’s flip this right here and say
so two is in the denominator of the denominator so let’s put 2 here and then
x minus 15 and so this is 4 over x minus 15 and
that is certainly not equal to x right there yeah that’s not an x right there
so the answer cannot be g right here a and of course if you’re only given two
choices you would have to say h of course maybe your answer is c none of
the above anyways let’s see if h is the right answer let’s just check that
so here we go let’s check the composition right here so here we go f

00:17
of and i’m going to use an h here for part b so f so this has to be true
in order to be in uh in it for them to be inverses of each other so let’s check
if this is true or not so i need to do work work work and i need to get to x
let’s see if that works out or not so h is going to go first so f of h of x
and i’m going to take what the h of x is and substitute it in right here so this
will be f of and what is the h 2 over x minus 5. so 2 over x minus five
and you know two over x and then all of that minus five
okay so now let’s take all of this wrap it up as one little thing and put it in
as an x so i’m going to get 2 over 2 over and then the input so all of this

00:18
so 2 over x minus 5 so 2 over all of that and then minus 5. so
here does all this simplify to an x and the answer is i don’t think so
because i have a minus five and a minus five so
i actually want this to be a positive don’t i if i could put a positive here
then i think all that will subtract off so let’s actually change this to a plus
five so let’s see if we can do that real quick let’s go with plus five
if i have a plus five here for my h now this h of x right here is going to
be two over x plus 5 so now i’m going to take 2 over x plus 5
and substitute it in and now i’m going to get 2 over
and then the input is going to be this right here all of that for the x

00:19
all right so now the fives add up to zero and i get two over two over x
in other words this is two over one in the numerator
times and then flip you get x over two so yeah you get x
so that works now hold on a second that may not be the inverse they may not be
inverse functions yet because what we haven’t checked is the
f composed h composed of f so what this one shows is that
whatever input you give to h f will undo that and get back to the x
but that doesn’t mean this one right here works so now i’m going to send the
x into f first and then we have to check that h will undo whatever the f does
so let’s see let’s send the x into f and we have a rule for that
so that is this all this right here so this will be h of 2 over x minus 5.
and so now i’m going to take all of this and plug it into the h and what does h

00:20
do 2 over the input so 2 over the input and all of this is the input
so two over input plus five so two over input plus five and now
uh does that simplify well let’s see think of this complex fraction right
here two let’s think about it as two over one times the reciprocal here
x minus five over two and then we still have a plus five
and so let’s go down here so the twos cancel so we’re going to get x minus 5
plus 5 and so yep we get an x so in fact these two functions right
here are inverses of each other f and h are inverses or

00:21
f and h are inverse functions all right so let’s um
look at finding some more just by not doing all this but just by
thinking about informally how to figure out what the inverse
function is so let’s try to do something like that
so let’s see the next example right here find the inverse of the function um
informally let’s think about it informally let’s get rid of this real quick
all right so here we go so part a i’m going to give you some functions here
now in an upcoming episode i’m going to show you exactly
step by step how to find the inverse but i think that before you actually try to
find that recipe to just kind of figure out how to do it
on your own how to think about it because you don’t want to think formally
every single time so i’ll give you a formal method um coming up here in

00:22
another episode with this episode let’s just try to do it informally and i think
that’s very instructive because you really want to have some kind of
intuition in terms about functions and inverses so if someone gives you f of x
equals 6x what do you think would be the inverse function
and by the way here’s the notation for inverse function so if your function is
named f we’ll use f with a minus 1 exponent but actually it’s not an
exponent it’s just notation that replaces that f is the inverse function of f
oops of f so so this right here f inverse is the inverse function of f okay so
what do you think would be the inverse function
what would be the undoing of multiplying something by 6
well i would think it would be dividing something by six i think these are

00:23
inverse functions of each other i think if you put something in here divided by
six and then send that over to here and multiply it by six
you’ll get back to where you started conversely if you start with something
over here and multiply it by 6 and then take that result and put it over here
and divide it by 6 i think you’ll get back to the x so informally these are
inverses of each other what about x plus six
what do you think would be the inverse function of x plus six
so we did one earlier where we said x plus four and the inverse we showed was
x minus four well it’s not going to be really any different adding 6 to
something how do you undo that you subtract 6 from something so if i give
an input i subtract 6. let’s try something like a combination
or how about let’s just do make sure we get it x minus three what would be the

00:24
inverse x plus three let’s try another one f of x is two x minus seven
f inverse is so here i’m going to uh put an input in and the first thing
i’m going to do is divi um sorry multiply by 2
and then i subtract 7. so we need to not only do undo those operations but we
need to undo them in the right order so here i’m multiplying by two first so
what i’m going to do last is divide by two
now after i multiply by two i’m going to subtract seven so what’s undoing
subtraction it’s adding so i’m going to add 7 first and then divide and this is
going to undo that this is going to multiply it by 2 and
then subtract 7. so these undo each other and then what do you think think about
if you have another power um like x to the third what do you think
would undo an x to the third power so if i send something in

00:25
as an input and i cube it i say like a four what’s four cubed four times four
times four so what would be the inverse it would be taking cube root of it
so these will undo each other this will raise it to the power three
this will take the root cube root and so yeah these will undo each other
so just informally these are some examples right here of
how you want to think about inverse function here all right so let’s do um
a one or two more examples here let’s go here and um
yep so we did those examples right there
and now let’s look at this example right here so let’s erase this right here and
let’s practice and making sure that we get this right here verifying an
inverse function because this has a technical meaning right here we say verify

00:26
so i’m given the function f and g and i say yeah they are inverses of each
other so i’m going to again i’m going to show you how to find the inverse in an
upcoming video but right now what we want to practice on is verifying in
other words someone told you hey those are inverse functions of each other and
you’re like uh let me verify if you’re right so i’m going to verify this right
here and see if we’re right so i need to check f composite of g of x
and i’m going to see if i get an x out of all that so here we go f of g of x
and then this will be the g of x will be this right here so it’ll be a minus and
then a 2x plus 6 over 7. so i’m going to take all of this right
here and substitute it in right there so i’m going to get minus 7 over 2
times the input which is all this so minus 7 over 2 times the x and then
minus 3. and let’s see what we get for all this

00:27
it looks like the 7s are going to cancel the minus 2s are going to cancel
well wait a minute there’s a 6 there isn’t there so what we need to do is to can
factor out the minus 2 don’t we so i’ll just write it like down here so
this is a minus 2x plus 6 over 7 is the same thing as minus 2 times 2
over 7 times and then what is it x minus 3 right
so we get minus two x and then plus six um
a minus needs to go for both so i think it’s going to be plus three actually
there we go so when i look at minus seven over two times this the
twos cancel the sevens cancel and so what we end up with is the x plus three

00:28
and then we still have the minus three there
and so then this turns out to be an x so just a little scratch work down here
hope hope it explains my thinking but basically the 7’s cancel it’s a minus
times a minus and the 2’s cancel but in order to cancel 2 you really need to
factor out a 2 first here so this will be 2 times x plus 3 and then we cancel
the twos but you still get an x plus 3 left all right so that’s just scratch
work there so there’s there’s an x right there and now to verify we need to
verify with f going first so this will be g of f of x
and so what is f of x it’s all of this right here so this would be minus seven
over two x and then -3 and so i need to take all of this
and send it to g so i need to put it there so i’m going to get a minus sign

00:29
out front and then the two times the whole input here
so two times the whole input plus six and then all of that but not the minus
sign so all of that gets over seven so let’s see can we simplify this
so the twos here cancel and so this will be a minus over seven
and so the twos cancel here well if i multiply the two times the
first one the twos will cancel and so i’m going to get minus 7x
and when i multiply the 2 times the minus 3 we’re going to get -6 right there
okay so and then we still get the plus six well
okay i almost forgot that part all right
so yeah the sixes add up to zero and the sevens cancel
and so this will be minus x and then we get a positive x out so i

00:30
didn’t touch that i left it right there and the 7s cancel leaving us with a
minus x right there all right so there we go there’s uh
f and g and that’s how we verify it right there if you’d like to see some
other examples let me know in the comments below and we
can work out some more examples um i hope you have a great day
and i look forward to seeing you in the next episode see you then
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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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