## SAT SUBJECT TEST MATH LEVEL 1

## FUNCTIONS

## CHAPTER 15

Functions and Graphs

• Relations

• Functions

• Combining Functions

• Composition of Functions

• Inverse Functions

• Exercises

• Answers Explained

**T**he concept of a function is one of the most fundamental notions in mathematics. In this chapter, you will see two equivalent definitions of *function* and then review the most important facts you need to know about functions for the Math 1 test.

### RELATIONS

Since a function is a special type of relation, we will first review the definition of a relation. A ** relation** is a set of ordered pairs. The first and second coordinates of the ordered pairs can be anything whatsoever, although on the Math 1 test they are almost always numbers. The number of ordered pairs in a relation can be finite or infinite. Each of the following sets is a relation:

*R*_{1} = {(1, 0), (1, 1), (3, 2)}

*R*_{2} = {(0, 1), (1, 1), (2, 1)} = {(*x*, *y*)| *x* = 0, 1, or 2 and *y* = 1}

*R*_{3} = {(0, 0), (1, 1), (2, 4)} = {(*x*, *y*)| *y* = *x*^{2} and *x* = 0, 1, or 2}

*R*_{4} = {(*x*, *y*)| *y* = *x* ^{2 }and *x* is an integer}

*R*_{5} = {(*x*, *y*)| *y* = *x*^{2 }and *x* 0}

*R*_{6} = {(*x*, *y*)| *y* = *x* ^{2}}

*R*_{7} = {(*x*, *y*)| *x* = *y* ^{2}}

*R*_{8} = {(*x*, *y*)| *x* ^{2 }+ *y* ^{2 }= 25}

*R*_{9} = {(*x*, *y*)| *x* is a state in the United States and *y* is the capital of *x*}

*R*_{10} = {(*x*, *y*)| *x* is a word in the English language and *y* is the number of letters in *x*}

*R*_{1}, *R*_{2}, *R*_{3}, *R*_{9}, and *R*_{10} are finite sets with 3, 3, 3, 50, and approximately 400,000 elements, respectively. *R*_{4}, *R*_{5}, *R*_{6}, *R*_{7}, and *R*_{8} are all infinite sets.

Note that (2, 4) is in *R*_{3}, *R*_{4}, *R*_{5}, and *R*_{6}; (–1, 1) is in *R*_{4} and *R*_{6}; (, 2) is in *R*_{5} and *R*_{6}; (4, 2) and (4, –2) are in *R*_{7}.

When the first and second coordinates of the ordered pairs are numbers, you can graph the pairs as explained in Chapter 13 on coordinate geometry. Here are the graphs of *R*_{1}, *R*_{2}, *R*_{3}, *R*_{4}, *R*_{5}, *R*_{6}, *R*_{7}, and *R*_{8}.