## SAT SUBJECT TEST MATH LEVEL 1

## FUNCTIONS

## CHAPTER 15

Functions and Graphs

### COMBINING FUNCTIONS

If *f* and *g* are two functions with overlapping—possibly equal—domains, then for all numbers *x* that are in both domains, it is possible to define the ** sum**,

**,**

*difference***, and**

*product***of**

*quotient**f*and

*g*.

**Key Fact N5**

**If x is in the domain of both f and g:**

• **( f** +

**=**

*g*)(*x*)**+**

*f*(*x*)

*g*(*x*)• **( f** –

**=**

*g*)(*x*)**–**

*f*(*x*)

*g*(*x*)• **( fg)(x)** =

**•**

*f*(*x*)

*g*(*x*)•

The only way KEY FACT N5 comes up on the Math 1 test is in evaluating the combination of two functions or in simplifying a quotient. If you need to evaluate the combination of functions, don”t actually combine them; use KEY FACT N5.

**EXAMPLE 10:** Let *f* (*x*) = 3*x* + 5 and *g* (*x*) = *x*^{2} + *x* – 5. To evaluate (*fg*)(2), you could first multiply *f* (*x*) • *g* (*x*):

*f* (*x*)*g*(*x*) = (3*x* + 5)(*x*^{2} + *x* – 5) = 3*x* ^{3} + 8*x*^{2} – 10*x* – 25

and then plug in *x* = 2:

3(2^{3}) + 8(2^{2}) – 10(2) – 25 = 24 + 32 – 20 – 25 = 11

but you shouldn”t. Instead you should evaluate first and then multiply:

(*fg*)(2) = *f* (2)*g*(2) = (3(2) + 5)(2^{2} + 2 – 5) = (11)(1) = 11.

**EXAMPLE 11:** If *f* (*x*) = (*x*^{2} + *x* – 2), *g* (*x*) = (*x*^{2} – *x* – 2), *h*(*x*) = (*x*^{2} – 1), and *x* 1, –1, then: