Lesson 6-3: Inverse Functions

Yesterday we got to push functions around a bit and see what we could do with them. Today we’re going to have some fun…we’re going to turn these things inside-out!

Turning a relation inside-out…the inverse of a relation

I wonder what a function looks like on the inside.  Today we are going to learn a new operation we can perform on a function. It is called the inverse of the function.

Now if you recall, a function is a special relation. Let’s first talk about what it means to find the inverse of a relation. Let’s play with the relation we used in lesson 6.1:

{ (0, 3), (1, 4), (1, 6), (2, -1), (3, 0) }

Getting the inverse of a relation is actually very simple: you just swap all the x’s with the y’s. Super easy! If you noticed, I colored the x values above with red and the y’s with blue. This will make it easier to see the inverse. And…here’s the inverse:

{ (3, 0), (4, 1), (6, 1), (-1, 2), (0, 3) }

That wasn’t bad was it? Let’s try it with an equation. Each equation represents a relation because it pairs x values with y values. So, how would we write the inverse of an equation?

Stumped? Okay, what does inverse mean again? That’s right, it means swap the x’s and the y’s. Hmm, how do we do that with an equation? Hey! Why don’t we just literally swap the x and y in the equation! Let’s try it with:

…now just switch the x and the y…we get

Hmm, how canbe the inverse of? Let’s check it out. Do a quick T-chart for each. If they are inverses of each other, then the T-charts should be identical except the x’s and y’s will be swapped in the 2nd T-chart.

I’m going to T-chart x = 0, 1 and 2 for the original equation. I’ll then take the y values from this T-chart and try them in the 2nd equation as the y’s:

x / y
0 / 1
1 / 3
2 / 5
x / y
1 / 0
3 / 1
5 / 2
x / y
? / 0
? / 1
? / 2

aaa

Yup, it works! Using the x values from the 1st as the y values in the 2nd shows the ordered pairs are swapped. These two functions really are inverses!

The inverse of a function

How do you take the inverse of a function? Here is one as an example:.

So what do we do? I don’t see a y so how in the world can I switch the x and the y? Wait a minute…isn’t a function an equation…it is just a special one isn’t it? Oh yeah! That means that sayingis the same as saying. Okay, I see now. We can just replacewith y then switch the x and y. Let’s try it:

…and the inverse is…

So that was pretty easy. 

Graphs of functions and their inverses

Let’s see what the graph of a function looks like compared to the graph of its inverse. We’ll use the function we just worked with:and its inverse.

Now to graph the inverse it will be easiest to convert it to slope intercept form:

Okay, now we can graph both equations…let’s put them both on the same graph:

The red line is the original functionand the yellow is its inverse. If you look at the dashed line as the reference line, you can see the graph of the inverse is the mirror reflection of the original function. Take a look at the point (0, 7) on the red line (original function). What is the inverse of that point? It is (7, 0). Look at the yellow line (inverse function)…it goes through (7, 0).

That line of reflection is the line y = x. It is the line of reflection for every function and its inverse.

One thing you can notice is that in this case, the inverse of the function is itself a function. There is no spot on the yellow line where a vertical line will intersects two points. This isn’t the case with all functions.

This is a bit easier to see with parabolas. Try it with. First find the inverse:. Now graph both equations using your graphing calculator. You should get get something like this:

Here the red is the line of reflection, the blue is the original function and the green is the inverse of the function. Is the inverse a function? Nope…you can see that a vertical line can intersect the green curve in two places. The inverse is not a function in this case.

To check if the inverse is a function, use your graphing calculator to graph it.

How to tell if two functions are inverses of each other

There is a quick trick you can use to check if two functions are inverses of each other: you use composition:

Ifare inverses of each other,

thenwill both equal x.

Check it out…verify thatare inverses of each other.

We took the composition of one with the other and then the other way around…and both times came up with x. This means that the two functions are inverses of each other!

Okay, try it again. Determine ifare inverses of each other.

Again, the composition of each with the other equals x so they are inverses.

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