Consider these two practice Quantitative problems:

1. Given

(A)

(B)

(C)

(D)

(E)

2. Given

(A)

(B)

(C)

(D)

(E)

3. If

(A)

(B)

(C)

(D)

(E)

If you find these questions completely incomprehensible, then you have found the right page.

The Quantitative section will ask an occasional question about function notation. Here is a basic catechism about functions and what you need to know about them for the QUANT.

A function is a rule, a “machine”, that takes an input and gives an output. When we are told the equation of a function, that equation makes explicit the rule this particular function is following. For example, for the function

Notice — this is a very subtle issue. The

When we write

Relatedly, the parentheses of function notation are mathematically inviolable. Nothing may pass through these parentheses. Again, this can be anti-intuitive, because when parentheses are used in ordinary notation, you can distribute through parenthesis, factor out, etc. Because a function is a different category of mathematical object, its parentheses are of a different nature. Thus

If you can simply avoid these mistakes and respect at all times the inviolability of the function’s parentheses, you will already be in better shape than a sizable portion of QUANT test takers.

In the above section, I discussed ways that folks new to functions might misinterpret function notation. Now, I am going to discuss how functions are seen by people who really understand them. Suppose we have the function

Where folks new to function just see the letter x, mathematicians see a “box”, an empty slot, a space that is, in some ways, analogous to an artist’s blank canvas. Anything that get plugged into the box on the left needs to get plugged into the box on the right. We can plug in numbers — any of the continuous infinity of real numbers on the real number line. We can also plug in algebraic expressions: If I put

If this is your first time encountering, or first time understanding, function notation, it is a worthwhile topic to practice, so that you are comfortable with it by test day. If you feel you have learned something from this, go back and try those two practice problems again before reading the solutions below.

1. We have the function

The right side says

Now, set those two equal and solve for

**Answer = B**

2. There are several ways to approach this problem. One quick way is to notice that if

Multiply both sides by the denominator.

Multiply both sides by

**answer = A**

3. First of all, see this Web Page and check the related lesson linked below for some background on function notation.

We can plug anything in for

So the notation

Remember that we have to square **both** the

All the answers are positive, so we choose

**answer = A**