SAT II Math II : Solving Piecewise and Recusive Functions

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

Example Question #1 : Solving Piecewise And Recusive Functions

Define  and  as follows:

Evaluate .

Possible Answers:

Correct answer:

Explanation:

 by definition.

 

 on the set , so 

.

 on the set , so

.

Example Question #65 : Functions And Graphs

Define function  as follows:

Give the range of .

Possible Answers:

Correct answer:

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 #2 : Solving Piecewise And Recusive Functions

Define function  as follows:

Give the range of .

Possible Answers:

Correct answer:

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 #3 : Solving Piecewise And Recusive Functions

Define functions  and  as follows:

Evaluate .

Possible Answers:

Undefined

Correct answer:

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 .

Possible Answers:

Undefined

Correct answer:

Undefined

Explanation:

First, evaluate  using the definition of  for :

Therefore, 

However,  is not in the domain of 

Therefore,  is an undefined quantity.

Example Question #69 : Functions And Graphs

Define functions  and  as follows:

Evaluate .

Possible Answers:

Undefined

Correct answer:

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 .

Possible Answers:

Undefined

Correct answer:

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 #6 : Solving Piecewise And Recusive Functions

Define two functions as follows:

Evaluate .

Possible Answers:

Correct answer:

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 #6 : 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:

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 

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