### All SAT II Math II Resources

## Example Questions

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

Define and as follows:

Evaluate .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

Undefined

**Correct answer:**

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

Since

, then

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

Define functions and as follows:

Evaluate Evaluate .

**Possible Answers:**

Undefined

**Correct answer:**

Undefined

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 .

**Possible Answers:**

Undefined

**Correct answer:**

First, evaluate using the definition of for :

Therefore,

Evaluate using the definition of for :

### Example Question #7 : Solving Piecewise And Recusive Functions

Define functions and as follows:

Evaluate .

**Possible Answers:**

Undefined

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 #9 : Solving Piecewise And Recusive Functions

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

**Possible Answers:**

None of these

**Correct answer:**

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

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