Functions

Functions andFunction Notation

100

Is the following relation a function? 

{(2, 4), (4, 4), (-1, 7)}

Yes

200

Given the function 

f(x)=sqrt(x+2

text{a. evaluate}\ f(7)

text{b. solve}\ f(x)=4

text{a.}\ +- 3

text{b.}\ x=14

300

Is the following function one-to-one? Explain.

No, because it fails the horizontal line test. 

400

Consider the relationship 

3r+2t=18

Write the relationship as a function 

r=f(t)

r=(18-2t)/3

r=-2/3t+6

Domain and Range

100

What is the domain of the following set?

{(2,4), (-4, 7), (-2, -1)}

{-4, -2, 2}

200

What is the domain and range of the following function in interval notation?

Domain

[-3, 2)

Range

[-5, 4]

300

Find the domain of the following function in interval notation:

(2x+1)/sqrt(5-x

(-oo, 5)

400

Does the following graph correspond to the piecewise function? Explain.

f(x)={(x^2,if x<=1),(3,if 1<x<=2):}`

No, we do not want to include the point (1, 3), so it should be an open circle. 

Absolute Value Functions

100

True or false:

The following function will have two solutions.

-|x+3|=9

False, it will have no solutions. 

200

True or False? Explain.

|X|>k\ text{is equivalent to} -k<X<k

|X|<k\ text{is equivalent to}\ X<-k or X>k

False, it should be 

|X|<k\ text{is equivalent to} -k<X<k

|X|>k\ text{is equivalent to} X<-k or X>k

300

Solve the following equation. 

3|x+1|-4=5

x=2 or x=-4

400

Solve the following inequality and write your answer in interval notation.

|3x-4|<=8

[-4/3, 4]

Composition of Functions

100

Are the following equivalent?

f(x)g(x) and f(g(x))

No, one is multiplication and the other is a composite function. 

200

text{Using the image below evaluate}\ g(f(0)).

5

300

text{Given}\ f(x)=2x^2+1 and g(x)=3x-5, text{find the following:}

f(g(2))

3

400

text{Write}\ f(x)=|x^2+7|\ text{as the composition of two functions,}

h(x)!=x and g(x)!=x text{, such that}\ f(x)=h(g(x)).

g(x)=x^2+7 and h(x)=|x|

or 

g(x)=x^2 and h(x)=|x+7|

Inverse Functions

100

True or false:

The input values of the original function will be the input values of the inverse function.

False, the input values of the original function will be the output values of the inverse function. 

200

text{How do we algebraically verify two functions}\ ,f(x) and g(x),

text{are inverses of each other?}

We need to check if 

f(g(x))=x and g(f(x))=x

300

Using the graph below, evaluate 

f^-1(0)

2

400

Find the inverse of the following function:

f(x)=3/(x-2)

f^-1(x)=3/y+2

Transformation of Functions

100

True or false:

Transformation values on the inside of the parenthesis and attached to the x lead to horizontal transformations. 

True

200

Write an equation for the graphed function below by using transformations of the graph of one of the toolkit functions. 

f(x)=sqrt(x+3

300

For the following, describe how the formula is a transformation of a toolkit function:

f(x)=sqrt(-2x+4

Horizontal reflection across the y-axis, horizontal compression by a factor of 1/2, and shifted right by 2 units. 

400

Write a formula for the function that results when the graph of the toolkit function is transformed as described:

text{The graph of}\ f(x)=1/x^2 text{is vertically compressed by a factor of}\ 1/3 

text{, then shifted to the left 2 units and down 3 units.}

1/(3(x+2)^2)-3

Rates of Change and Behavior of Functions

100

What is the average rate of change formula?

(\Deltay)/(\Deltax)

(f(b)-f(a))/(b-a)

200

True or False:

text{The following function is increasing on the interval}\ [1, oo).

text{False, the function is increasing on the interval}\ (1, oo).

300

For the following graph, identity all absolute extrema. 

Absolute max is 16 when x = -2 and x = 2 (2, 16) and (-2,16)

Absolute min is -10 when x=3 (3, -10)

400

For the following, find the average rate of change on the interval specified. 

h(x)=x^2\ text{on}\ [1,5]

6