SAT II Math II : Graphing Piecewise and Recusive Functions

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

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Example Questions

Example Question #1 : Graphing Piecewise And Recusive Functions

Define a function  as follows:

How many -intercept(s) does the graph of  have?

Possible Answers:

Two

Four

None

One 

Three

Correct answer:

None

Explanation:

To find the -coordinates of possible -intercepts, set each of the expressions in the definition equal to 0, making sure that the solution is on the interval on which  is so defined.

 on the interval 

 or 

However, neither value is in the interval , so neither is an -intercept.

 

 on the interval 

However, this value is not in the interval , so this is not an -intercept.

 

  on the interval 

However, this value is not in the interval , so this is not an -intercept.

 

 on the interval 

However, neither value is in the interval , so neither is an -intercept.

 

The graph of  has no -intercepts.

Example Question #1 : Graphing Piecewise And Recusive Functions

Define a function  as follows:

How many -intercept(s) does the graph of  have?

 

Possible Answers:

Four

One

None

Three

Two

Correct answer:

Two

Explanation:

To find the -coordinates of possible -intercepts, set each of the expressions in the definition equal to 0, making sure that the solution is on the interval on which  is so defined.

 

 on the interval 

However, this value is not in the interval , so this is not an -intercept.

 

 on the interval 

 or 

 is on the interval , so  is an -intercept. 

 

 

 on the interval 

 is on the interval , so  is an -intercept. 

 

  on the interval 

However, this value is not in the interval , so this is not an -intercept.

 

The graph has two -intercepts,  and .

Example Question #2 : Graphing Piecewise And Recusive Functions

Define function  as follows:

Give the -intercept of the graph of the function.

Possible Answers:

The graph does not have a -intercept.

Correct answer:

Explanation:

To find the -intercept, evaluate  using the definition of  on the interval that includes the value 0. Since 

on the interval  ,

evaluate:

The -intercept is .

Example Question #3 : Graphing Piecewise And Recusive Functions

Define a function  as follows:

At which of the following values of  is  discontinuous?

I) 

II) 

III) 

Possible Answers:

I and II only

None of I, II, and III

I and III only

All of I, II, and III

II and III only

Correct answer:

I and III only

Explanation:

To determine whether  is continuous at , we examine the definitions of  on both sides of , and evaluate both for :

 

 evaluated for :

 evaluated for :

Since the values do not coincide,  is discontinuous at .

 

We do the same thing with the other two boundary values 0 and .

 

 evaluated for :

 evaluated for :

Since the values coincide,  is continuous at .

 

 turns out to be undefined for , (since  is undefined), so  is discontinuous at .

 

The correct response is I and III only.

Example Question #4 : Graphing Piecewise And Recusive Functions

Define a function  as follows:

At which of the following values of  is the graph of  discontinuous?

I) 

II) 

III) 

Possible Answers:

I and III only

I and II only

None of I, II, and III

All of I, II, and III

II and III only

Correct answer:

II and III only

Explanation:

To determine whether  is continuous at , we examine the definitions of  on both sides of , and evaluate both for :

 

 evaluated for :

 evaluated for :

Since the values coincide, the graph of   is continuous at .

 

We do the same thing with the other two boundary values 0 and 1:

 

 evaluated for :

 evaluated for :

Since the values do not coincide, the graph of  is discontinuous at .

 

 evaluated for :

 evaluate for :

Since the values do not coincide, the graph of  is discontinuous at .

 

II and III only is the correct response.

 

Example Question #5 : Graphing Piecewise And Recusive Functions

Define a function  as follows:

Give the -intercept of the graph of the function.

Possible Answers:

The graph does not have a -intercept.

Correct answer:

Explanation:

To find the -intercept, evaluate  using the definition of  on the interval that includes the value 0. Since 

on the interval  ,

evaluate:

The -intercept is .

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