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Math 104: Calculus14 chapters | 114 lessons | 9 flashcard sets

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Lesson Transcript

Instructor:
*Robert Egan*

If you use a function to map a to b, is there a way to go back from b to a again? Learn how to find and graph inverse functions so that you can turn a into b and back into a.

If I tell you that I have a function that maps the number of feet in some distance to the number of inches in that distance, you might tell me that the function is *y*=*f(x)* where the input *x* is the number of feet and the output *y* is the number of inches. You might even tell me that *y* = *f(x)* = 12*x*, because there are 12 inches in every foot. But what if I told you that I wanted a function that does the exact opposite? What if I want a function to take the number of inches as input and return the number of feet as output? Could you tell me what this function is?

**Inverse functions** are exactly that. If we have a function *y*=*f(x)*, then the inverse function is written as *y*= *f*^(-1)(*x*), and it does the exact opposite of the function. What happens if you put a function and its inverse into a **composite** function such as *f*^(-1)(*f(x)*)? First, we evaluate the inner function, *f(x)*, then we're going to evaluate the outer function *f*^(-1)(*x*).

Let's take a look at an example. Say we start with 4 feet. Well, our function is *f(x)*=12*x* because there are 12 inches in every foot. If we plug in 4 feet to start, then f(4) = 12 * 4 = 48 inches. Now if we take the inverse function, and the inverse function is going to be *f*^(-1)(*x*) = *x*(1/12). So, if we take 48 inches, then our inverse function, f^(-1)(48) = 48 / 12 = 4 feet. Okay, so you might be able to find *f(x)* and *f*^(-1)(*x*) just based on your understanding of inches and feet, but how do you do it in general?

- Write your function out in terms of
*x*and*y*:*y*=*f(x)*. - Swap the
*x*and*y*variables:*x*=*f(y)*. - Solve for
*y*as a function of*x*. - Set
*y*=*f*^(-1)(*x*). - Check the composite function:
*f*^(-1)(*f(x)*).

Following these steps, let's say we have a function *f(x)* = 3(*x* - 1) + 2.

We're going to write this out in terms of *x* and *y*: *y* = 3(*x* - 1) + 2. Then we're going swap the *x* and *y* variables, so we're going to write this as *x* = 3(*y* - 1) + 2. This can be a confusing step if you're not careful, but at its heart, all you're doing is putting *x* everywhere you see *y* and putting *y* everywhere you see *x*. Then you're going to solve for *y* as a function of *x*. So I'm going to subtract 2 from both sides, *x* - 2 = 3(*y* - 1), divide both sides by 3, (*x* - 2) / 3 = *y* - 1 and add 1 to both sides and I end up with *y* = 1 + (*x* - 2)/3.

I'm going to call what's on the right-hand side my inverse function, f^(-1)(*x*) = 1 + (*x*-2)/3. Finally, I'm going to check my answer, so I'm going to find f^(-1) of (*f(x)*). To do this, I'm going to write *f(x)* = 3(*x*-1) + 2. I'm going to plug that in as input for my inverse function, so f^(-1)(*x*) = 1 + ((3(*x*-1) + 2) - 2)/3. I have my input here, so I'm just going to solve and simplify for f^(-1)(*x*) = 1 + (3(*x*-1))/3: f^(-1)(*x*) = 1 + *x* - 1. And sure enough, f^(-1)(*f(x)*) = *x*, which is exactly what we'd expect.

So what about a function like y = round(*x*)? Remember that round(*x*) just rounds our input to the nearest integer: round(4.2) = 4. However, round(4.8) = 5 and round(5.1) = 5. In this case, do you think that you can find an inverse function that can take 5 and give your either 5.1 or 4.8? No, round(*x*) is a function that has no inverse.

What about the function *f(x)* = *x*^3 + 3*x*? I can write it out in terms of *x* and *y*: *y* = *x*^3 + 3*x*. I can then swap the variables, *x* = *y*^3 + 3*y*. I can then solve it for *y* - but that's not immediately obvious to me. Is there another way? Let's go back and look at an easier function, like *f(x)* = 3*x* - 6. I end up with a graph that looks like this, a simple line. Now I'm going to graph the inverse, which is f^(-1)(*x*) = (*x* + 6)/3. So the inverse is this blue line; it looks a lot like the original function, except it's mirrored. And it's actually mirrored over the 45-degree angle, which is the *x*=*y* line. If I could fold this paper in half, then I'd see that the function and its inverse become the same line. I can use this on much more complex functions too. Say I was looking at a function like this. If I draw the 45-degree line and mirror it, then I can get a pretty good idea of what that inverse function looks like.

The **inverse function** will undo the function. That means that the inverse function of the function will give you back what you started with. But not all functions will have inverses. For example, *y*= round(*x*) doesn't have an inverse. You can find the inverse function with our five-step process. If you graph a function and its inverse, they're 45-degree reflections of one another. That's an easy way to find the inverse or get an idea of what the inverse function looks like for really complex functions.

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Math 104: Calculus14 chapters | 114 lessons | 9 flashcard sets

- What is a Function: Basics and Key Terms 7:57
- Graphing Basic Functions 8:01
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
- Polynomials Functions: Exponentials and Simplifying 7:45
- Exponentials, Logarithms & the Natural Log 8:36
- Slopes and Tangents on a Graph 10:05
- Equation of a Line Using Point-Slope Formula 9:27
- Horizontal and Vertical Asymptotes 7:47
- Implicit Functions 4:30
- Go to Graphing and Functions

- Go to Continuity

- Go to Limits

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