# College Algebra : Miscellaneous Functions

## Example Questions

### Example Question #1 : Miscellaneous Functions

Define a function .

Which statement correctly gives ?

Possible Answers:

Correct answer:

Explanation:

The inverse function  of a function  can be found as follows:

Replace  with :

Switch the positions of  and :

or

Solve for . This can be done as follows:

Square both sides:

Add 9 to both sides:

Multiply both sides by , distributing on the right:

Replace  with :

### Example Question #1 : Miscellaneous Functions

Refer to the above diagram, which shows the graph of a function .

True or false: .

Possible Answers:

True

False

Correct answer:

False

Explanation:

The statement is false. Look for the point on the graph of  with -coordinate  by going right  unit, then moving up and noting the -value, as follows:

, so the statement is false.

### Example Question #1 : Miscellaneous Functions

The above diagram shows the graph of function  on the coordinate axes. True or false: The -intercept of the graph is

Possible Answers:

True

False

Correct answer:

False

Explanation:

The -intercept of the graph of a function is the point at which it intersects the -axis (the vertical axis). That point is marked on the diagram below:

The point is about one and three-fourths units above the origin, making the coordinates of the -intercept .

### Example Question #171 : College Algebra

A function  is defined on the domain  according to the above table.

Define a function . Which of the following values is not in the range of the function ?

Possible Answers:

Correct answer:

Explanation:

This is the composition of two functions. By definition, .  To find the range of , we need to find the values of this function for each value in the domain of . Since , this is equivalent to evaluating  for each value in the range of , as follows:

Range value: 3

Range value: 5

Range value: 8

Range value: 13

Range value: 21

The range of  on the set of range values of  - and consequently, the range of  - is the set . Of  the five choices, only 45 does not appear in this set; this is the correct choice.

### Example Question #1 : Miscellaneous Functions

Evaluate:

Possible Answers:

Correct answer:

Explanation:

Evaluate the expression  for , then add the four numbers:

### Example Question #1 : Miscellaneous Functions

Evaluate:

Possible Answers:

Correct answer:

Explanation:

Evaluate the expression  for , then add the five numbers:

### Example Question #1 : Miscellaneous Functions

refers to the floor of , the greatest integer less than or equal to .

refers to the ceiling of , the least integer greater than or equal to .

Define and

Which of the following is equal to ?

Possible Answers:

Correct answer:

Explanation:

, so, first, evaluate  by substitution:

, so evaluate by substitution.

,

the correct response.

### Example Question #1 : Miscellaneous Functions

refers to the floor of , the greatest integer less than or equal to .

refers to the ceiling of , the least integer greater than or equal to .

Define and .

Evaluate

Possible Answers:

Correct answer:

Explanation:

, so first, evaluate using substitution:

, so evaluate using substitution:

,

the correct response.

### Example Question #1 : Miscellaneous Functions

Consider the polynomial

,

where  is a real constant. For  to be a zero of this polynomial, what must  be?

Possible Answers:

None of the other choices gives the correct response.

Correct answer:

Explanation:

By the Factor Theorem,  is a zero of a polynomial  if and only if . Here, , so evaluate the polynomial, in terms of , for  by substituting 2 for :

Set this equal to 0: