## SAT SUBJECT TEST MATH LEVEL 1

## FUNCTIONS

## CHAPTER 15

Functions and Graphs

### COMPOSITION OF FUNCTIONS

A more important way to combine two functions is to form their ** composition**:

*f*o

*g*.

The definition of composition is given in KEY FACT N6.

**Key Fact N6**

**The composition of two functions f and g is (f o g)(x) = f (g(x)).**

The composition of *f* and *g* can be formed only if some of the numbers in the range of *g* are in the domain of *f*.

**EXAMPLE 12:** If *f*(*x*) = and *g* (*x*) = –*x*^{2} – 2, then (*f o g*)(*x*) does not exist. For every real number *x*, *g* (*x*) is negative, but no negative number is in the domain of *f*.

**EXAMPLE 13:** Let *f* (*x*) = 2*x* + 3 and *g* (*x*) = *x*^{2} – 1.

To find (*f o g*)(3) you have two choices:

**(1)** You can determine *f* (*g* (*x*)) and then plug in *x* = 3:

(*f o g*)(*x*) = *f* (*g*(*x*)) = 2(*g* (*x*)) + 3 = 2(*x*^{2} – 1) + 3 = 2*x*^{2} – 2 + 3 = 2*x*^{2} + 1

Then (*f o g*)(3) = 2(3)^{2} + 1 = 2(9) + 1 = 19

or

**(2)** You can calculate (*f o g*)(3) directly:

(*f o g*)(3) = *f* (*g* (3)) = *f* (8) = 2(8) + 3 = 19

**EXAMPLE 14:** Let *f* and *g* be the same functions as in Example 13: *f* (*x*) = 2*x* + 3 and *g* (*x*) = *x*^{2} – 1. Then:

Of course, (*g o f* )(3) could have been calculated directly:

(*g o f* )(3) = *g* (*f* (3)) = *g* (9) = 9^{2} – 1 = 80

Notice that, in general, (*f o g*) (*g o f* ). In Examples 13 and 14, you saw:

(*f o g*)(3) = 19 and (*g o f* )(3) = 80

It is possible, however, to have functions *f* and *g* for which (*f o g*) = (*g o f* ).

**EXAMPLE 15:** Let *f* (*x*) = 3*x* – 2 and Then: