Algebra II : Functions as Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #11 : Introduction To Functions

Function

The above table refers to a function  with domain .

Is this function even, odd, or neither?

Possible Answers:

Odd

Even

Neither

Correct answer:

Odd

Explanation:

A function is odd if and only if, for every  in its domain, ; it is even if and only if, for every  in its domain, . We can see that

It follows that  is an odd function.

Example Question #12 : Functions And Graphs

Function

The above table refers to a function  with domain .

Is this function even, odd, or neither?

Possible Answers:

Even 

Neither

Odd

Correct answer:

Neither

Explanation:

A function is odd if and only if, for every  in its domain, ; it is even if and only if, for every  in its domain, . We can see that 

;

the function cannot be even. This does allow for the function to be odd. However, if  is odd, then, by definition, 

, or

and  is equal to its own opposite - the only such number is 0, so

.

This is not the case -  - so the function is not odd either.

Example Question #13 : Functions And Graphs

Function

The above table refers to a function  with domain .

Is this function even, odd, or neither?

Possible Answers:

Neither

Even

Odd

Correct answer:

Even

Explanation:

A function  is odd if and only if, for every  in its domain, ; it is even if and only if, for every  in its domain, . We can see that 

Of course, 

.

Therefore,  is even by definition.

Example Question #14 : Functions And Graphs

Odd

Which of the following is true of the relation graphed above?

Possible Answers:

It is an even function

It is not a function

It is a function, but it is neither even nor odd.

It is an odd function

Correct answer:

It is an odd function

Explanation:

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

 Odd

Also, it can be seen to be symmetrical about the origin. Consequently, for each  in the domain,  - the function is odd.

Example Question #15 : Functions And Graphs

Which is a vertical asymptote of the graph of the function  ?

(a) 

(b) 

Possible Answers:

(a) only

Both (a) and (b)

(b) only

Neither (a) nor (b)

Correct answer:

(a) only

Explanation:

The vertical asymptote(s) of the graph of a rational function such as  can be found by evaluating the zeroes of the denominator after the rational expression is reduced. The expression is in simplest form, so set the denominator equal to 0 and solve for :

The graph of  has the line of the equation  as its only vertical asymptote.

Example Question #16 : Functions And Graphs

Which of the following is a vertical asymptote of the graph of the function  ?

(a)

(b)  

Possible Answers:

(a) only

Both (a) and (b)

Neither (a) nor (b)

(b) only

Correct answer:

Neither (a) nor (b)

Explanation:

The vertical asymptote(s) of the graph of a rational function such as  can be found by evaluating the zeroes of the denominator after the rational expression is reduced

First, factor the numerator. It is a quadratic trinomial with lead term , so look to "reverse-FOIL" it as

by finding two integers with sum  and product 30. By trial and error, these integers can be found to be  and , so

Therefore,  can be rewritten as 

.

Cancelling , this can be seen to be essentially a polynomial function:

,

which does not have a vertical asymptote.

Example Question #17 : Functions And Graphs

True or false: The graph of  has as a horizontal asymptote the graph of the equation .

Possible Answers:

True

False

Correct answer:

True

Explanation:

 is a rational function whose numerator and denominator have the same degree (1). As such, it has as a horizontal asymptote the line of the equation , where  is the quotient of the coefficients of the highest-degree terms of its numerator and denominator. Consequently, the horizontal asymptote of 

is 

,

or

.

Example Question #18 : Functions And Graphs

True or false: The graph of  has as a horizontal asymptote the graph of the equation .

Possible Answers:

True

False

Correct answer:

True

Explanation:

 is a rational function whose numerator and denominator have the same degree (2). As such, it has as a horizontal asymptote the line of the equation , where  is the quotient of the coefficients of the highest-degree terms of its numerator and denominator. Consequently, the horizontal asymptote of 

is

or

.

Example Question #19 : Functions And Graphs

Even

Which of the following is true of the relation graphed above?

Possible Answers:

It is an odd function

It is not a function

It is a function, but it is neither even nor odd.

It is an even function

Correct answer:

It is an even function

Explanation:

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as seen below:

Relation

Also, it is seen to be symmetrical about the -axis. This proves the function even.

Example Question #20 : Functions And Graphs

Even

Which of the following is true of the relation graphed above?

Possible Answers:

It is not a function

It is an even function

It is a function, but it is neither even nor odd.

It is an odd function

Correct answer:

It is an even function

Explanation:

The relation graphed is a function, as it passes the vertical line test - no vertical line can pass through it more than once, as is demonstrated below:

Relation

Also, it is seen to be symmetrical about the -axis. Consequently, for each  in the domain,  - the function is even.

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