### All SAT II Math II Resources

## 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:**

None

Two

Three

One

Four

**Correct answer:**

None

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 #2 : Graphing Piecewise And Recusive Functions

Define a function as follows:

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

**Possible Answers:**

Two

Three

None

One

Four

**Correct answer:**

Two

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 #3 : 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:**

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 #4 : Graphing Piecewise And Recusive Functions

Define a function as follows:

At which of the following values of is discontinuous?

I)

II)

III)

**Possible Answers:**

II and III only

All of I, II, and III

I and II only

None of I, II, and III

I and III only

**Correct answer:**

I and III only

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 #5 : 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 II only

I and III only

All of I, II, and III

II and III only

None of I, II, and III

**Correct answer:**

II and III only

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 #6 : 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:**

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 .