Inverse functions

An overview of the inverse of a one-to-one function.

>> Hi, again. This is Tim Hsu at San Jose State University. And I'm here today

to talk about inverse functions. I'll start by talking about what a one to one

function is. We'll talk a little bit about the definition of the inverse of a

one to one function. We will compare the global attributes of a function f and

its inverse f inverse. And finally, even though, as you'll see, this is not

strictly necessary for calculus, we'll talk about finding the formula for an

inverse function. Okay, so first we've got to talk about one to one functions

and what they are. So a one to one function, y equals f of x is a function that

never takes the same y value twice, it never has the same output twice. So

equivalently another way of describing one to one functions, is that a function

y equals f of x is one to one exactly when if and only if any horizontal line

that passes through the graph of y equals f of x it passes through it at most

once. You may know that as the horizontal line test. All right, so turning to

some examples of functions, here's an example. The sine function graphed here

from negative pi to positive pi, so the sine function is not one to one because

it takes the value of 1/2, at least twice over here and over here you can see

it. And you can see that it fails the horizontal line test because this red

horizontal line here at height 1/2 passes through the graph twice, fail. So the

sine function is not one to one. On the other hand, it can be shown that the

function f of x equals 2 to the x is one to one because it never attains the

same line value twice, so whatever kind of horizontal line you have here only

ever passes the graph once. Now, that's not obvious from what we talked about so

far, but it is true. For the future, let me recall, that the domain of 2 to the

x is all real values of x and the range of 2 to the x is all y zero. Okay, so

now we turn to the definition of the inverse of one to one functions and one

example, which we just saw. So suppose y equals f of x is a one to one function.

Then the inverse function, f inverse of x, is defined by saying that y is equal

to f inverse of x if and only if f of y equals x. In other words, here's a sort

of more verbal and less equationy way of looking at this. F inverse of x is

defined to be the input that you would need to give the forward function in

order to get x as an output, right. So again, once again, so what is f inverse

of x? It's the input you need to give the forward function to get x as an output

of the forward function. So and I'm fussing over the definition here and sort of

going over it a bit here because that is what you use in practice. Out of all

the things we talk about in this video, the most useful thing is actually this

theoretical nonsense, the definition of the inverse. Okay, so here's one

important consequence of the definition of f inverse that follows that, f

inverse of f of x is equal to x, right. Because what is f inverse? It's the

thing that you would need to feed f in order to get the input the output f of x,

well, that's just exactly x. And for all x and the domain of f and similarly, f

of f inverse of x is also equal to x, for all x in the domain of f inverse. So

this is the point of f inverse, this is the reason why we define inverse

functions. The point of an inverse function is that it undoes what f does, and

vice versa. So to look at a concrete example, so let f of x equal 2 to the x. As

it happens, this is a one to one function, which is again, not obvious but true.

So it has an inverse. So let's look at the question. What is f inverse of 32?

Very concrete question. Answer, well, by definition, as we were saying before, f

inverse of 32 is the input that you need to give the forward function f of x

equals 2 to the x to get an output of 32. So I want you to think about like what

x you could plug in to get an output of 32 and that will be the answer. Pause

your video now. Okay, in other words, f inverse of 32 is the solution to f of x

equals 32, or, in other words, f inverse of 32 is the solution to 2 to the ? Is

equal to 32. And again, see if you can pause the video now and think about what

should go in there to make that equation work. Well, hopefully you realized that

since f of 5 equals 2 to the 5 plus 32, you pause the video one more time, and

try to figure out what that tells you about f inverse of 32. Well, what that

tells you is precisely that f inverse of 32 is 5. So let's turn our attention to

more global matters. How do f and f inverse compare overall? How do they look in

the big picture? Well, the sort answer to that is that they look exactly the

same except everything that happens with f, you switch x and y and you get

something that happens f inverse. So for example, the domain of f becomes a

range of f inverse and the range of f becomes the domain of f inverse. The graph

of f inverse is actually the graph of f but you flip it over or mirror it by the

line y equals x. Okay, so returning to our example where the forward function is

f of x equals 2 to the x, again, let me repeat that the domain of 2 to the x is

all real x, and the range of 2 to the x is all y greater than zero. But on the

other hand, if here we have the graph of the inverse function, so here's a graph

of the forward function, f of x equals 2 to the x. And here's a graph of the

inverse function, f inverse of x. So if you take my word for it that is in fact

the inverse function of f of x, we see that the domain of the inverse function

is all positive x, just as the range of the forward function was all y greater

than zero. And the range of the inverse function is all real numbers. That's a

little harder to see. That corresponds with the fact that the domain of the

forward function is all real numbers. And furthermore, as we were saying before,

the graph of the inverse function, here, is the graph of f of x flipped over

this sort of line slope 1 through the origin, line y equals x. Here's the

forward function, you flip it over, y equals x to get the inverse, the graph of

the inverse function. Okay, so finally, we turn to finding the formula of an

inverse function. Let me start with a hard truth, which is that, if you're just

watching this for the purpose of calculus, feel free to skip to the end. Because

calculus pretty much only really needs inverse functions when we don't have

other formulas for them. So the most prominent example is the exponential

functions, the whole reason for talking about inverse functions with respect to

exponential functions, is that we don't have a better formula for the inverse

function 2 to the x to the natural log of 2, which is just another way of saying

the inverse function of 2 to the x. However, just for completeness, to find the

formula of f inverse, here's what you do. Given a formula for f, especially an

algebraic formula, one, you set up the equation y equals f of x. You switch the

variables x and y, step two. Three, you solve for y. And four, the end result is

that you get something that looks like y equals some stuff and x, and the answer

f inverse of x is equal to that stuff in x. All right, so turning to an example.

So take f of x equals x cubed minus 7, turns out that's not obvious but it's

true, that one to one. So to find a formula for x inverse of x, you set y equals

x cubed minus 7 and switch the variables x and y to get that x is equal to y

cubed minus 7. And then you solve for y. so then you add 7 to both sides and get

x plus 7 equals y cubed. Take the cube to both sides and get that y is equal to

the cube root of x plus 7. And so that's your answer, f inverse of x is equal to

the cube root of x plus 7. All right, well, what are you doing watching this? Go

do something useful instead. Thanks very much and bye.