# High School: Functions

### Building Functions HSF-BF.B.4

4. Find inverse functions.

A lot of relationships are one way streets. Your students are probably facing that harsh reality in the form of unrequited love. They can buy bouquets of flowers and a heart-shaped boxes of chocolates, but it doesn't mean they'll get any back. Too bad they already ordered that enormous stuffed lovebug.

One-way relationships might be unavoidable in life, but math is much kinder (and your students should remember that). In mathematics, we can have relationships that are two-way streets. They're called inverse functions, and students should know how to find them.

Easier than finding love, anyway.

#### Drills

1. What is the inverse function of f(x) = 3x + 7?

The first order of business is to replace f(x) with y and then to switch y and x so that we have x = 3y + 7. Now, we can rearrange the equation to solve for y, which gives us . Since (A) doesn't include 7 in the numerator, and (C) and (D) add 7 rather than subtract it, the only right answer is (B).

2. What is the inverse function of

If we exchange f(x) with y, and then switch the x with the y, we'll have . Solving for y gives us the inverse function, given by (C). This takes basic algebra and very little else.

3. What is the inverse function of f(x) = 2x?

f-1(x) = log2 x

If we go through the normal ritual to find the inverse, we'll have x = 2y. The only way to get y on the ground floor and out of the exponent penthouse is to use the logarithm. Understanding what logarithms are helps tremendously, since we know that a = bc means logb a = c. That means we have y = log2 x, or (C). Taking the log of both sides would have also given us the same answer.

4. Which two functions are inverses of each other?

f(x) = 4x – 5 and

In order to determine whether one function is the inverse of another, we can just plug them into one another. So if we solve f(g(x)) for (A), (B), and (C), we should be able to figure out which two functions are inverses of each other. For (A), we end up with f(g(x)) = x, so they are inverses. The other two, on the other hand, are not since f(g(x)) = 0.5x for (C) and (B) wants us to divide by 0! The only right answer has to be (A).

5. Which two functions are inverses of each other?

and

If f(g(x)) = x, then f(x) and g(x) are inverses of each other. In order to find the two inverse functions, all we need to do is solve each answer choice for f(g(x)) and see if it equals x. Hopefully we know outright that (A) is not true, which also means (D) isn't right, either. While (B) gives us f(g(x)) = x, (C) results in , which is most certainly not x.

6. A linear function has the points (-1, 2), (0, 3), and (2, 5). Which of the following points will the inverse function have?

(7, 4)

Since this function is described as "linear," we know that the x and y values will keep changing at the same slope. We could even go as far as finding the equation that describes the line (y = x + 3). The inverse function will have points that are the same as the original function, but with the x and y coordinates switched. So we know that it will have (2, -1), (3, 0), and (5, 2). None of those points are listed and (A) and (B) can be automatically ruled out because they directly conflict with (3, 0) and (5, 2). But what about (7, 4)? We can see that the original function will result in the coordinates (4, 7), which means the inverse function will have a point with coordinates (7, 4).

7. A linear function has the points (-1, 3), (-1, 4), and (-2, 4). Which of the following points will the inverse function have?

None of the above

Why is (D) the right answer when the point (-1, 3) is listed clear as crystal? After all, an inverse function will have all the points of the original function with the x and y coordinates switched, right? While that's true, the question lied to you. It said that we have a linear function, but how is that possible with the points (-1, 3) and (-1, 4)? A function means only one output for every input, and this has two outputs. That means whatever relation we have here is not a function. There will be no inverse function, and there will be no switching of x and y coordinates. Even if we did, switch the coordinates, we couldn't have (4, -1) and (4, 2) because that relation isn't a function, either. Sorry if we messed you up on that one.

8. Which of the following graphs represents the graph of f-1(x) if f(x) = 0.5x + 2?

We could solve this problem in many different ways, all of which should give us (C) as the right answer. Plotting the points of f(x), switching the coordinates, and comparing them to the graphs should point to (C). We could also plot the actual function on each of the answer graphs to see which answer is a reflected image of the original function across the y = x line. Either way, we should end up with (C) regardless.

9. In order to make the function f(x) = x2 + 6x + 9 an invertible function, its domain should be which of the following?

x ≥ -3

We can factor the function so that f(x) = (x + 3)2 which means that it intersects the x-axis at its line of symmetry, x = -3. The domain of the original function correlates to the range of the inverse function. Since our inverse function is f-1(x) =  - 3, and there's no value of x that will make f-1(x) less than -3, that means the domain of our original function should be all values greater than or equal to -3, or (A).

10. In order to make the function f(x) = |x| an invertible function, its domain should be which of the following?