## SAT SUBJECT TEST MATH LEVEL 1

## FUNCTIONS

## CHAPTER 15

Functions and Graphs

### INVERSE FUNCTIONS

If *f* and *g* are functions such that

(1) for every *x* in the domain of *g*, *f* (*g* (*x*)) = *x* and

(2) for every *x* in the domain of *f*, *g* (*f* (*x*)) = *x*,

then we say that *g* is the ** inverse** of

*f*and write

*g*=

*f*

^{–1}, which is read “

*f*inverse.” It is also true that

*f*is the inverse of

*g*:

*f*=

*g*

^{–1}.

**Key Fact N7**

**If for some function f, f**

^{–1}

**exists, then**

*f* (*f*^{–1}**( x))** =

*x*and*f*^{–1}

**(**=

*f*(*x*))

*x*.The inverse, *f* ^{–1}, of a function, *f*, undoes what *f* does. In Example 15, *f* multiplies a number by 3 and then subtracts 2 from it; *g*, which is *f* ^{–1}, adds 2 to a number and then divides the result by 3.

Not every function has an inverse, but many do. On the Math 1 test, you may be asked to find the inverse of a particular function. The procedure to do this is given in KEY FACT N8.

**Key Fact N8**

**If f is a function of x, to find f**

^{–1}

**, first write**=

*y*

*f*(*x*). Then interchange*x*and*y*and solve for*y*.**EXAMPLE 16:** If *f* (*x*) = 3*x* – 2, what is *f* ^{–1}(*x*)?

Write *y* = 3*x* – 2. Then switch the *x* and *y*: *x* = 3*y* – 2. Now solve for *y* :

So *f* ^{–1}*x* = . Note that *f* ^{–1}, which is function *g* in Example 15, simply undoes what *f* does: *f* multiplies a number by 3 and then subtracts 2; *f* ^{–1} adds 2 to a number and then divides the result by 3.