# High School: Functions

### Building Functions HSF-BF.B.5

5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Students should have pieced together the puzzle by now, more or less. The inverse of adding is subtracting. The inverse of multiplying is dividing. The inverse of squaring is square-rooting (and the same goes for any circle-roots or triangle-roots). Now it's time for the final piece of the puzzle: the inverse of exponentiating is logarithmifying (which may or may not be a real word).

Students should already know that an exponential equation a = bc can be rewritten in logarithmic form as c = logba or

They may not have used logarithms in a while, but it should still be there, under the piles of memorized Jonas Brothers lyrics and Gangnam Style dance moves. After they've yanked it back out and dusted it off, tell them why it's useful.

Students should be able to use this exponent-logarithm relationship when finding the inverse of a function. For instance, we have the function f(x) = 3x, which we can treat as y = 3x. Finding the inverse means switching x and y, and then solving for y. So what we have is really x = 3y. We can now use the wonderful world of logarithms to solve for y. We should get y = log3x ≈ 2.1logx.

If your students aren't convinced that these functions are inverses, do whatever you need to do to prove it. You can graph them and show the line of symmetry, plot points and switch them, or calculate f(f-1(x)) and prove that it equals x. All of those should be enough evidence to support the fact that exponentials and logarithms are inverses.

#### Drills

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

f-1(x) = log2x

The first order of business is to replace f(x) with y and then to switch y and x so that we have x = 2y. Using our knowledge of logarithms, we can rearrange this into y = log2x. This means (B) is the right answer.

2. What is the inverse function of f(x) = 5x + 3?

If we exchange f(x) with y, and then switch the x with the y, we'll have x = 5y + 3. Solving for y gives us the inverse function. We should get x – 3 = 5y, and we can then take the log of both sides. As long as we do it properly, we should end up with log(x – 3) = y log 5, or . That means the right answer is (B).

3. What is the inverse function of f(x) = 3 × 7x – 2?

If we go through the normal ritual to find the inverse, we'll have x = 3 × 7y – 2. The first thing we should do is divide both sides by 3 to get . Now, we can change this into logarithmic form, which is , which translates to (A). Answer (B) switches the 3 and the 7, while (C) and (D) add 2 before it should actually be added.

4. Which two functions are inverses of each other?

All of the above

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. We could also just find the inverses of f(x) and see if f-1(x) = g(x). We end up with f(g(x)) = x or f-1(x) = g(x) for (A), (B), and (C). They're all inverses of each other and (D) is the right answer.

5. Which two functions are inverses of each other?

f(x) =10x and g(x) = log x

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. If we do the math, we can see that (A) and (C) are not right, while (B) looks as though it's missing something. It doesn't have a base or a denominator, but it's actually correct. A logarithm without any listed base is assumed to be 10, so (B) is the right answer.

6. An exponential function has the points (0, 1), (1, 9), and (2, 81). Which of the following points will the inverse function have?

(729, 3)

Since this function is described as "exponential," we know that x is the exponent of some base. We could even go as far as finding the equation that describes the function (y = 9x). 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 (81, 2), (9, 1), and (1, 0). None of those points are listed and (B) can be automatically ruled out because it directly conflicts with (1, 0) and (9, 1). Since the inverse function is a logarithm, a negative x is impossible, but (A) is not. We can see that the original function will result in the coordinates (3, 729), which means the inverse function will have a point with coordinates (729, 3).

7. An exponential function has the points (0, 1), (1, 3), (2, 9), (3, 27). Which of the following points will the inverse function have?

(9, 2)

An inverse function will have all the points of the original function with the x and y coordinates switched. The exponential function here is y = 3x. We know that if we switch the x and y coordinates, we'll have (1, 0), (3, 1), (9, 2), and (27, 3). Of our answer choices, (A) applies to the inverse function because it switches the points.

8. Which of the following graphs represents the graph of f-1(x) if f(x) = 1.2x?

We could solve this problem in many different ways, all of which should give us (A) as the right answer. Plotting the points of f(x), switching the coordinates, and comparing them to the graphs should point to (A). 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 (A) regardless.

9. What is the domain for an exponential function?

x can be all real numbers

If any old "exponential function" is too abstract to think about, we can take the function y = 2x as an example. The numbers we can input for x are limitless. They don't need to be only positive or only negative, and we can substitute x = 0 as well. So (A) is the right answer.

10. What is the domain of a logarithmic function?

x > 0