**Functions**

Think of a function as a machine: You put a value in, and a value comes out. The function machine is also consistent. Thus, if you put in a specific value, like, say, 8, then a value comes out that depends only on 8. If your function machine produces 5 when you input 8, then it will always produce 5 when you input 8.

This page gives some simple examples of functions. The first example shows a function named f, which is defined by a quadratic polynomial. The second example show the composition of two functions, named f and g. And the third example talks about a sequence, which is a function whose domain is a set of integers.

**To practice more of these types of problems, click** here.

**Example 1)**

If f(x) = −3x² − 5x + 11, then what is f(−5)?

A. −47

B. −43

C. – 39

D. −35

E. −33

**Explanation: **To solve this problem, we’ll substitute -5 for x:

f(x) = -3x² – 5x + 11

f(-5) = -3(-5)² – 5(-5) + 11

= (-3 × 25) + 25 + 11

= -75 + 25 + 11

= -39

**Answer C is correct.**

**Example 2)**

If f(x) = -3x – 9 and g(x) = 3x + 23, then what is f(g(-9))?

A. 2

B. 3

C. 4

D. 5

E. 6

**Explanation: **We’ll start with the inside part of the question and work our way out. We need to first determine what g(-9) is, so we’ll substitute the -9 for x in the g(x) function.

g(x) = 3x + 23

g(-9) = 3(-9) + 23

= -27 + 23

= -4

We’ll now substitute the -4 for x in the f(x) function.

f(x) = -3x – 9

f(-4) = -3(-4) – 9

= 12 – 9

= 3

**So B is the correct answer.**

**Example 3)**

If a sequence starts at 17, and ends at -4, with each number being 3 less than the previous number, how many numbers are in the sequence?

**Explanation: **The difference between the first term, 17, and the last term, – 4, is:

17 – (-4) = 21

And since each term is 3 different from the previous term, then there must be 21/3 = 7 terms from 17 to -4.

But that doesn’t include the first number in the sequence, 17. So there are actually 7 + 1 = 8 numbers in the sequence.

**The correct answer is 8.**

**To practice more of these types of problems, click** here.