# SAT II Math II : Solving Piecewise and Recusive Functions

## Example Questions

### Example Question #1 : Solving Piecewise And Recusive Functions

Define  and  as follows:

Evaluate .

Explanation:

by definition.

on the set , so

.

on the set , so

.

### Example Question #2 : Solving Piecewise And Recusive Functions

Define function  as follows:

Give the range of .

Explanation:

The range of a piecewise function is the union of the ranges of the individual pieces, so we examine both of these pieces.

If , then . To find the range of  on the interval , we note:

The range of this portion of  is .

If , then . To find the range of  on the interval , we note:

The range of this portion of  is

The union of these two sets is , so this is the range of  over its entire domain.

### Example Question #3 : Solving Piecewise And Recusive Functions

Define function  as follows:

Give the range of .

Explanation:

The range of a piecewise function is the union of the ranges of the individual pieces, so we examine both of these pieces.

If , then

To find the range of  on the interval , we note:

The range of  on  is .

If , then

To find the range of  on the interval , we note:

The range of  on  is .

The range of  on its entire domain is the union of these sets, or .

### Example Question #4 : Solving Piecewise And Recusive Functions

Define functions  and  as follows:

Evaluate .

Undefined

Explanation:

First, we evaluate  . Since , the definition of  for  is used, and

Since

, then

### Example Question #4 : Solving Piecewise And Recusive Functions

Define functions  and  as follows:

Evaluate Evaluate .

Undefined

Undefined

Explanation:

First, evaluate  using the definition of  for :

Therefore,

However,  is not in the domain of

Therefore,  is an undefined quantity.

### Example Question #6 : Solving Piecewise And Recusive Functions

Define functions  and  as follows:

Evaluate .

Undefined

Explanation:

First, evaluate  using the definition of  for :

Therefore,

Evaluate  using the definition of  for :

### Example Question #5 : Solving Piecewise And Recusive Functions

Define functions  and  as follows:

Evaluate .

Undefined

Explanation:

First we evaluate . Since , we use the definition of  for the values in the range :

Therefore,

Since , we use the definition of  for the range :

### Example Question #8 : Solving Piecewise And Recusive Functions

Define two functions as follows:

Evaluate .

Explanation:

By definition,

First, evaluate , using the definition of  for nonnegative values of . Substituting  for 5:

; evaluate this using the definition of  for nonnegative values of :

12 is the correct value.

### Example Question #21 : Solving Functions

Which of the following would be a valid alternative definition for the provided function?

None of these

Explanation:

The absolute value of an expression  is defined as follows:

for

for

Therefore,

if and only if

.

Solving this condition for :

Therefore,  for

Similarly,

for

The correct response is therefore