A very useful cousin of a function is

called the function’s inverse. Do you remember a special type of

function called a one-to-one function? Well, what made it special? First of all,

every element in the domain was paired with a unique element in the range. That’s what made it a function, period. What

makes it a one to one function is that it works the other way as well. It’s a special type of function that

every element in the range is paired with a unique element in the domain. In other words it works backwards. Every

X is paired with only one y and every y is paired with only one x. Now

while a function is basically a path, if you would, to get from x to y, the inverse of a function is the

way to get back. I say, to get back from y, back to X. Now

consider the reason we’re talking about one-to-one functions is unless you’re

working with a one-to-one function that works both ways, if you would, there isn’t a way back. So you only have

an inverse of a function or a function only has an inverse if it is in fact

originally a one-to-one function. Otherwise, there’s no way back. Anyway, now this relation, is a function; it gets the function

seal of approval. because no x is paired with

more than one y. The inverse of it would be just flipping

the domain and the range. It’s very simple but that’s the idea of

the inverse. The inverse is going to map the range back to the domain or the y’s

back to the x’s. Now this isn’t always that simple. Let’s look at this relation. Now

every X is paired with only one y. So it is a function; it does get the function seal of

approval. All the x’s are in fact different. If we tried to find the inverse of it,

basically by flipping the X’s and the Y’s, Well, in this case, all the x’s aren’t

different. So, while it would have been an inverse,

it’s not an inverse function and the reason for that, the inverse isn’t a

function is because the original function wasn’t a one-to-one function,

was it? It was a function but it wasn’t a one-to-one function. So we’ll have to

start with a one-to-one function in the first place to even have an inverse ever.

Here is how we find the inverse formula of a

function. There’s four steps. A lot of this is just

notation changes. For instance, the first step is to replace the f of X with what

it really is the Y. Then we’re actually going to interchange the x and the y in

the equation. Flip them and then here’s something you should be good at… We’re

going to solve for y. Once we’ve solved for y and we’ve gotten

y by itself, we’re going to replace that y with the

notation for F inverse of X and that’s written f minus 1 of X. It doesn’t mean

to the negative 1 power. It means the inverse function of X. [I don’t understand. ] Ok, well, let’s try one.

We’re going to try and find the inverse of this function f of x. The first thing

we’ll do is replace f of x with y. Okay, there you go. Second step,

interchange the X and the y so I’ll write the function normally except I’ll

interchange the x and y. Ok, that’s the second step. Now the third

step is to do whatever it takes to solve for y; in other words get y by itself. Now, who’s keeping y from being by itself?

If you look at him, the four is. So I’ll subtract 4 from both sides and y will equal x minus 4. That’s basically the inverse function.

But for notation sake the fourth step is to replace the y part

with the notation for that F inverse of x and that’s the inverse. That’s the

inverse function of X in this case. Recall that the original function will

take any X that you put ,in any value for x and pair it with a new y. I would

like to say map every X to its own unique y. The

inverse function is going to get you back from that y, back to x. Interestingly enough you can test

whether you’re correct, whether you’ve found the correct inverse function by

doing something that we just covered called compositions. Ok, if we compose the inverse function

with the original function or f inverse of f of x it should bring us back to x.

and let’s try that. [Let’s play here we go.] We’re going to find the composition of

the inverse of f of X. Ok so we’re going to take the inverse of the original

function. Now the original, I’m sorry, the inverse function is X minus 4 so we’re

going to replace X minus four, the X portion of that, with the original

function. I’d like to use parentheses here and now

I’ll simplify and look what I end up getting back to the mother lode x that’s that’s how I’ll know. I check if

I can compose f inverse with f of X and I get back to X. Well, then I know I have

the right inverse and cool as a mule. You know what, if we actually graph any

function and on the same picture, graph its inverse. Iwant to show you what

happens when we graph the original function. I’ll get three easy points. We plug in 0,

1 and negative 1 because they’ll fit on the graph. I put in a 0, I get four. I put

in a 1, I get five and if I put a negative 1, I get three. I graph these three points. They line up

and there’s the original function, y or f of x equals x plus 4. Now if I

do the inverse function, try the same three points, 0 negative 0, 1 and negative

1, I get (0,negative 4), (1, negative 3) and (negative 1, negative 5) and if I graph

those, they also line up and the inverse function in blue their graphs right

there Now, what will always happen when you

graph a function and it’s inverse function since one is the, if you would,

interchanging of x and y is that they will be a mirror of each other about the

line y equals x. I could actually fold this graph on the

dotted line which is that the graph of the line y equals x and the red would

end up on top of the blue. That will always happen. Ok so the graphs of inverse functions

always mirror each other about the line y equals x. That’s another test that you could use.

On your calculator, even, let’s try another. We’re going to find the inverse of this

function, the function f of x equals 3x plus 2. We need to find out what function would

get us back from y back to x. I, well, remember the steps! Replace f of X

with Y because that’s what it really is and then rewrite the whole equation

interchanging x and y. Ok, now solve the equation for y; get y by

itself. Now I have two things keeping y from

being by itself, the two and the three. Let me subtract

two from both sides and I’ll get x minus 2 there. Can’t, can’t

do anything there. Now divide both sides by 3 to undo the 3 and y will equal x

minus 2 over 3. Now the last step is to replace the Y

with the notation F inverse of X. This doesn’t surprise me much that the

inverse of multiplying by 3 and adding to the inverse of that is subtracting 2

and dividing by 3. It’s just the opposite operations, isn’t

it? So there’s your inverse function. Now let’s test it out not because I said so but just to

practice with compositions. Remember that if we compose each we should end up back

at x. The composition of f , uh, inverse of x and f of x should get me back to x. Let’s try it. There’s the inverse

function and I’m going to replace X with the original function 3x plus 2. Now let me simplify. The twos cancel and

the threes cancel and what’s left? There he is, x. We got back to X. I must

have found the correct inverse function Pretty cool. Let’s do the other test. Let’s graph both the original function

and it’s inverse and see if it mirrors about the line y equals x. The original

function put in these three values and I’ll get these three points. Now it’s a

little bit different than our picture before but they do line up. Now the

inverse function and we get three points for him and let’s see me graph those

points, hmm… looks different. They do line up and, son of a gun, if you draw the line y equals x and if you fold it on that dotted line that red would end

up on top of that blue. So the graphs of inverse functions

always mirror each other about the line y equals x meaning that you could fold

your graph over the line y equals x on the dotted line and they fall on top of

each other. Ok, let’s go practice it. Go do that

homework.

yeah i agree with his comment the audios are kinda random dont put in too much if not any at all

but other than that very helpful thanks

2:33 star trek audio lol

Thank you so much 😀

Thank You so much i left my notes at school and this video helped me so much! Thank You i really cant say it enough you made it so simple to understand 🙂

This video made a boring subject kinda fun!

Thank you so easy to learn

math with fun thankyou vryyyyyyyyyyyyyyyyyyyyyyyyy mch

thank u so much…your video is simple and easy

this is the best explanation on the net

What it the funtion inverse for f (x)=x?

Two words For the marker of this math video: Thank you!

easy to understand by students.

1000thanks dear fellow!

this guy is even better than that hot chick mathbff lol…… i finally paid attention in class today and after i come to youtube to clarify what i didnt understand from my broken english math teacher.

can u answer me

if f(x+3)=3x+4. find f(x)

Thank you so much you explained it in the most simplest form everyone else makes it all complicated

We just out here tryna function!!

Thanks to your videos I will pass my finals!!

love the random sounds c:

I love the video! You are one of the teacher needed in Community College. It took me more then a week to understand my teacher, and just 12 minute to get this! huge difference! Omg Thank you so much for made this video!

great explanation! 🙂

I FINALLY understand!!!! Thanks Bill!!!

Understandable and clear explanation

When YouTube teaches you better than your teacher thanks man!

can you make a video with inverse functions containing fractions?

those sound effects tho

Thanks dude, this video is more vivid than my teacher's lecture! Hope i can pass my quiz!

I was wondering if you had a video on how to find the domain and range of the one-to-one function defined by f(x) and the inverse

A BIG HELP Thank You very much!!!

Thank you sooooooo much for this video. I understand.

THANK YOU THANK YOU THANK YOU!! This helped me a lot. I can't say "thank yous" enough but thank you!!

Very easy explanations, how to find a inverse of function? …impressed me much.

What a fantastic explanation 👌🏾👌🏾👌🏾👌🏾

thanks a

TONwhen you do checking, the answer is always X ?

please reply ASAP…

Thank You !!! I get it now 😂

my math teacher only covers it very few times and next thing you know we have a test. and next thing you know I'm failing but this has helped Alot. I understand it more

thanks

just AWESOME….THANX MAN….GREAT CREATIVITY

I have a question. When you graphed the two different sets of functions and inverse functions, you didn't switch the domains and ranges in the little T chart (or x and f(x) chart) for both, only the second example. In the first example you kept the domain the same for the function and the inverse function, but interchanged them in the second example. Can you explain why you did this and/or if that was intentional?

wow thank you for taking the time to make such a clear and to the point video, your content is one of the best out there believe it or not, please keep on doing what you are doing, also what made you decide to do this.

I really like how you read and answer comments on the same day, or a day after☺

we're watching this in class lol

thanks this was helpful

Very nice

Thanks brosky 👌

Best video I've seen on the topic. Thank you.

really you did it as how every one can understand thanks

thanks god you're a lifesaver,my teacher is a witch and she didn't teach me this at all

f(x)=x^2-3x+5 we have to find inverse first bit i cant and find also f inverse (15)

very good video

This would be good without the autistic sound effects that are distracting

life time movie

thank you, this helped me a lot. We haven't discussed this but I understand it now.

Boi

Sry gotta switch tutorial, cant focus with these soundeffects.

The key for an effective powerpoint is to use as little distractions as possible.

But then again, its a 6 year old video

First comment in 3 years, also those sound effects tho

That’s very nice