Functional Math Problems, Part II

In our last post, we looked at some questions involving functions, specifically functions involving money. Today, we’re going to continue our discussion with a look at how functions can be manipulated.

First of all, remember that evaluating functions involves simply plugging information in. For example, if f(x) = x + 1, then f(2) = 2 + 1 = 3.

The rules of function math are pretty simple: Treat the function like you would any other variable. Let’s jump in with a question:

Let the function h be defined by h(x) = 5x + 2. If 2h(√t) = 24, what is the value of t?

(A)   4
(B)   9
(C)   12
(D)   16
(E)    25

This may look somewhat complicated, but just remember that working with functions is similar to working with regular algebraic equations. Let’s take this question step by step.

We have our function h(x) = 5x + 2. Pretty straightforward. The difficulty arises with the second part of the question, which states that 2h(√t) = 24. Where do we start? Both sides are easily divisible by 2, so let's divide:

2h(√t) = 24

h(√t) = 12

Note that we treated h(√t) just like any other variable in an equation-solving question. Now we need to look at our function h(x). In this case, x = √t. This means that any x in our function can be replaced by √t. Plugging this into the function, we get:

h(x) = 5x + 2

h(√t) = 5√t + 2.

Now all we have left to do is solve for t. We know that h(√t) = 12 from above, and we also have the equation for h(√t):

h(√t) = 5√t + 2 = 12

Subtract 2 from both sides:

5√t = 10

Divide both sides by 5:

√t = 2

Square both sides:

(√t)² = 2²

t = 4.

We have our answer: t = 4. Bubble in answer (A).

Let’s take a look at another function question:

A certain function p has the property that p(x*y) = p(x) + p(y). Which of the following must then be true?

I. p(x²) = 2p(x)
II. p(2xy) = 2p(x) + p(y)
III. p(3xy) = p(x) + p(3y)

(A)   I only
(B)   II only
(C)   I and II
(D)   I and III
(E)    I, II, and III

These types of questions are somewhat difficult and may take some time to complete. Keep in mind that all we need to do is to determine which of the three statements are always true, given the information we have. This question asks us to assume that a certain “property” is true for our function. We are given a function p that has the property p(x*y) = p(x) + p(y). For example, for p(12), we could say:

p(12) = p(6*2) = p(6) + p(2).

Luckily, we don’t need to worry about calculating anything. All we need to determine is, given this property for our function, which of these three statements is always true. The best way to do this is make each statement resemble our given property. If we can show that these statements are really just another way of representing that property (which we know is true), then we'll know that the statement must be true as well.

Statement I: p(x²) = 2p(x). We know that x² is simply x*x, so this can be written as:

p(x*x) = 2p(x)

or

p(x*x) = p(x) + p(x).

Instead of x*y in our example, we just have x*x. This will always be true for our function, given this property. Statement I is always true. We can rule out answer choice (B).

Statement II: p(2xy) = 2p(x) + p(y). This one is a little more complicated. We have 3 terms (2, x, y) as a part of our function. Where do we start?

It looks like the right side of the statement is trying to divide 2xy into 2x and y in some sort of manner. Let’s try to do the same:

p(2xy) = p(2x*y)

According to our function’s property:

p(2x*y) = p(2x) + p(y).

Is this the same as what statement II is saying? Not really. The “p(y)” portion is the same, but is p(2x) = 2p(x)? No, not necessarily.

Statement II is not always true, so (C) and (E) cannot be our answers. We’re down to either (A) or (D). Let's see if statement III is true.

Statement III: p(3xy) = p(x) + p(3y). This looks similar to statement II, but it's not quite the same. Again, we have three terms (3, x, y) as a part of our function. The right side of the statement appears to be dividing 3xy into x*3y, so let’s try:

p(3xy) = p(x*3y)

According to the property:

p(x*3y) = p(x) + p(3y).

That’s exactly what statement III says. Statement III is always true. Our answer must be (D).

One benefit to these "which of these statements is true" questions is that you can usually make an educated guess. Let's say that you figured out for sure that statements I and III are true, but you're not so sure about statement II. Given what you know, you can eliminate all but two answer options. Even if you guess, you've still got a 50% chance of getting it right.

Here are some key points for function math questions:

• Functions are based on simple plugging-in. If f(x) = x + 1, f(2) = 2 + 1. It’s really that simple. Even something crazy like f(%!$#) is just equal to %!$# + 1.
• Adding, multiplying, subtracting, and dividing functions is the same as with any other variable. As you saw in our first question, performing operations on functions is the same as in typical equation solving. If f(x) = x + 1, 2f(x) is just 2(x+1) or 2x + 2.

And here are some keys for the multiple statement questions:

• Make your statements resemble the information you're given. If you can show that a statement is essentially saying the same thing as the given property, you should be able to find your answer.
• Make an educated guess. If you know one or two statements are true or false, you can narrow down your choices. Don't be afraid to make an educated guess if you can eliminate some answers.

Take another look at function math. And try this quiz for some additional practice.

Got an issue with math? Share in the comments, and send your questions to testpreptutor@sparknotes.com.

Related Post: Money Problems: Equation Solving Edition

← Newer Posts | Older Posts →