Introduction to Functions

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Questions 1 - 10
1

Which analysis can be performed to determine if an equation is a function?

Vertical line test

Horizontal line test

Calculating zeroes

Calculating domain and range

Explanation

The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one (or ) value for each value of . The vertical line test determines how many (or ) values are present for each value of . If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.

The horizontal line test can be used to determine if a function is one-to-one, that is, if only one value exists for each (or ) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.

Example of a function:

Example of an equation that is not a function:

2

Which analysis can be performed to determine if an equation is a function?

Vertical line test

Horizontal line test

Calculating zeroes

Calculating domain and range

Explanation

The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one (or ) value for each value of . The vertical line test determines how many (or ) values are present for each value of . If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.

The horizontal line test can be used to determine if a function is one-to-one, that is, if only one value exists for each (or ) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.

Example of a function:

Example of an equation that is not a function:

3

Evaluate if and .

Undefined

Explanation

This expression is the same as saying "take the answer of and plug it into ."

First, we need to find . We do this by plugging in for in .

Now we take this answer and plug it into .

We can find the value of by replacing with .

This is our final answer.

4

Question

If the function is depicted here, which answer choice graphs ?

A

D

C

B

None of these graphs are correct.

Explanation

The function shifts a function f(x) units to the left. Conversely, shifts a function f(x) units to the right. In this question, we are translating the graph two units to the left.

To translate along the y-axis, we use the function or .

5

Evaluate if and .

Undefined

Explanation

This expression is the same as saying "take the answer of and plug it into ."

First, we need to find . We do this by plugging in for in .

Now we take this answer and plug it into .

We can find the value of by replacing with .

This is our final answer.

6

Question

If the function is depicted here, which answer choice graphs ?

A

D

C

B

None of these graphs are correct.

Explanation

The function shifts a function f(x) units to the left. Conversely, shifts a function f(x) units to the right. In this question, we are translating the graph two units to the left.

To translate along the y-axis, we use the function or .

7

Let and . What is ?

Explanation

THe notation is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).

The original expression for f(x) is . We will take each x and substitute in the value of g(x), which is 2x-1.

We will now distribute the -2 to the 2x - 1.

We must FOIL the term, because .

Now we collect like terms. Combine the terms with just an x.

Combine constants.

The answer is .

8

Let and . What is ?

Explanation

THe notation is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).

The original expression for f(x) is . We will take each x and substitute in the value of g(x), which is 2x-1.

We will now distribute the -2 to the 2x - 1.

We must FOIL the term, because .

Now we collect like terms. Combine the terms with just an x.

Combine constants.

The answer is .

9

Which of the following does NOT belong to the domain of the function f(x)=\frac{x^{-1}+\sqrt{1-x}}{4x^2+1} ?

0

-1

-1/2

1/2

1

Explanation

The domain of a function includes all of the values of x for which f(x) is real and defined. In other words, if we put a value of x into the function, and we get a result that isn't real or is undefined, then that value won't be in the domain.

If we let x = 0, then we will be forced to evaluate 0^{-1}, which is equal to 1/0. The value of 1/0 is not defined, because we can never have zero in a denominator. Thus , because f(0) isn't defined, 0 cannot be in the domain of f(x).

The answer is 0.

10

If , which of these values of is NOT in the domain of this equation?

Explanation

Using as the input () value for this equation generates an output () value that contradicts the stated condition of .

Therefore is not a valid value for and not in the equation's domain:

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