ORCCA Functions Chapter Review

Section 11.6 Functions Chapter Review

Subsection 11.6.1 Function Basics

In Section 1 we defined functions informally, as well as function notation. We saw functions in four forms: verbal descriptions, formulas, graphs and tables.

Example 11.6.1. Informal Definition of a Function.

Determine whether each example below describes a function.

The area of a circle given its radius.
The number you square to get $9\text{.}$

Explanation.

The area of a circle given its radius is a function because there is a set of steps or a formula that changes the radius into the area of the circle. We could write $A(r)=\pi r^2\text{.}$
The number you square to get $9$ is not a function because the process we would apply to get the result does not give a single answer. There are two different answers, $-3$ and $3\text{.}$ A function must give a single output for a given input.

Example 11.6.2. Tables and Graphs.

Make a table and a graph of the function $f\text{,}$ where $f(x)=x^2\text{.}$

Explanation.

First we will set up a table with negative and positive inputs and calculate the function values. The values are shown in Figure 3, which in turn gives us the graph in Figure 4.

input, $x$	output, $\operatorname{f}(x)$
$-3$	$9$
$-2$	$4$
$-1$	$1$
$0$	$0$
$1$	$2$
$2$	$4$
$3$	$9$

Figure 11.6.3.

a graph of the points from the table connected with a dashed line — Figure 11.6.4. $y=f(x)=x^2$

Example 11.6.5. Translating between Four Descriptions of the Same Function.

Consider a function $f$ that triples its input and then adds $4\text{.}$ Translate this verbal description of $f$ into a table, a graph, and a formula.

Explanation.

To make a table for $f\text{,}$ we’ll have to select some input $x$-values so we will choose some small negative and positive values that are easy to work with. Given the verbal description, we should be able to compute a column of output values. Table 6 is one possible table that we might end up with.

$x$	$f(x)$
$-2$	$3(-2)+4=-2$
$-1$	$3(-1)+4=1$
$0 $	$3(0)+4=4$
$1 $	$3(1)+4=7$
$2 $	$3(2)+4=10$

Figure 11.6.6.

Once we have a table for $f\text{,}$ we can make a graph for $f$ as in Figure 7, using the table to plot points.

Lastly, we must find a formula for $f\text{.}$ This means we need to write an algebraic expression that says the same thing about $f$ as the verbal description, the table, and the graph. For this example, we can focus on the verbal description. Since $f$ takes its input, triples it, and adds $4\text{,}$ we have the formula

\begin{equation*} f(x)=3x+4\text{.} \end{equation*}

a graph of the points listed in the table; the points are connected with a dotted line and form a parabola — Figure 11.6.7. $y=f(x)$

Subsection 11.6.2 Domain and Range

In Section 2 we saw the definition of domain and range, and three types of domain restrictions. We also learned how to write the domain and range in interval and set-builder notation.

Example 11.6.8.

Determine the domain of $p\text{,}$ where $p(x)=\dfrac{x}{2x-1}\text{.}$

Explanation.

This is an example of the first type of domain restriction, when you have a variable in the denominator. The denominator cannot equal $0$ so a bad value for $x$ would be when

\begin{align*} 2x-1\amp=0\\ 2x\amp=1\\ x\amp=\frac{1}{2} \end{align*}

The domain is all real numbers except $\frac{1}{2}\text{.}$

Example 11.6.9.

What is the domain of the function $C\text{,}$ where $C(x)=\sqrt{2x-3}-5\text{?}$

Explanation.

This is an example of the second type of domain restriction where the value inside the radical cannot be negative. So the good values for $x$ would be when

\begin{align*} 2x-3\amp\geq0\\ 2x\geq 3\\ x\geq \frac{3}{2} \end{align*}

So on a number line, if we wanted to picture the domain of $C\text{,}$ we would make a sketch like:

$a number line, with a bracket at 3/2 that opens to the right, with an arrow pointing to the right$

The domain is the interval $\left[\frac{3}{2},\infty\right)\text{.}$

Example 11.6.10. Range.

Find the range of the function $q$ using its graph shown in Figure 11.

$A parabola that opens downward is pictured with the range indicated as an interval on the y-axis; there is a bracket at 1 and an arrow that points downward along the y-axis.$

Figure 11.6.11. $y=q(x)\text{.}$ The range is marked as an interval on the $y$-axis.

Explanation.

The range is the collection of possible numbers that $q$ can give for output. Figure 11 displays a graph of $q\text{,}$ with the range shown as an interval on the $y$-axis.

The output values are the $y$-coordinates so we can see that the $y$-values start from $1$ and continue downward forever. Therefore the range is $(-\infty,1]\text{.}$

Subsection 11.6.3 Using Technology to Explore Functions

In Section 3 we covered how to find a good graphing window and use it to identify all of the key features of a function. We also learned how to solve equations and inequalities using a graph. Here are some examples for review.

Example 11.6.12. Finding an Appropriate Window.

Graph the function $t\text{,}$ where $t(x)=(x+10)^2-15\text{,}$ using technology and find a good viewing window.

Explanation.

A graph of the function in a viewing window of -7 to 7 on the x and y axes. The graph is barely visible on the left side of the graph. This is not a good viewing window. — Figure 11.6.13. $y=t(x)$ in the viewing window of $-7$ to $7$ on the $x$ and $y$ axes. We need to zoom out and move our window to the left.

A graph of the function with the window dimensions expanded and shifted to the left. Now we see it is a parabola with the vertex and intercepts all in the window. — Figure 11.6.14. $y=t(x)$ in a good viewing window.

After some trial and error we found this window that goes from $-20$ to $2$ on the $x$-axis and $-20$ to $100$ on the $y$-axis.

Now we can see the vertex and all of the intercepts and we will identify them in the next example.

Example 11.6.15. Using Technology to Determine Key Features of a Graph.

Use the previous graph in figure 14 to identify the intercepts, minimum or maximum function value, and the domain and range of the function $t\text{,}$ where $t(x)=(x+10)^2-15\text{.}$

Explanation.

From our graph we can now identify the vertex at $(-10,-15)\text{,}$ the $y$-intercept at $(0,85)\text{,}$ and the $x$-intercepts at approximately $(-13.9,0)$ and $(-6.13,0)\text{.}$

A graph of the function with the window dimensions expanded and shifted to the left. The window goes from. Now we see it is a parabola and the vertex and intercepts are in the window. — Figure 11.6.16. $y=t(x)=(x+10)^2-15\text{.}$

Example 11.6.17. Solving Equations Graphically Using Technology.

Use graphing technology to solve the equation $t(x)=40\text{,}$ where $t(x)=(x+10)^2-15\text{.}$

Explanation.

To solve the equation $t(x)=40\text{,}$ we need to graph $y=t(x)$ and $y=40$ on the same axes and find the $x$-values where they intersect.

From the graph we can see that the intersection points are approximately $(-17.4,40)$ and $(-2.58,40)\text{.}$ The solution set is $\{-17.4,-2.58\}\text{.}$

A graph of the same function in the expanded window along with the horizontal line y=40. — Figure 11.6.18. $y=t(x)$ where $t(x)=(x+10)^2-15$ and $y=40\text{.}$

Subsection 11.6.4 Simplifying Expressions with Function Notation

In Section 4 we learned about the difference between $f(-x)$ and $-f(x)$ and how to simplify them. We also learned how to simplify other changes to the input and output like $f(3x)$ and $\frac{1}{3}f(x)\text{.}$ Here are some examples.

Example 11.6.19. Negative Signs in and out of Function Notation.

Find and simplify a formula for $f(-x)$ and $-f(x)\text{,}$ where $f(x)=-3x^2-7x+1\text{.}$

Explanation.

To find $f(-x)\text{,}$ we use an input of $-x$ in our function $f$ and simplify to get:

\begin{align*} f(\substitute{-x})\amp=-3(\substitute{-x})^2-7(\substitute{-x})+1\\ \amp=-3x^2+7x+1 \end{align*}

To find $-f(x)\text{,}$ we take the opposite of the function $f$ and simplify to get:

\begin{align*} \highlight{-}f(x)\amp=\highlight{-}(-3x^2-7x+1)\\ \amp=3x^2+7x-1 \end{align*}

Example 11.6.20. Other Nontrivial Simplifications.

If $g(x)=2x^2-3x-5\text{,}$ find and simplify a formula for $g(x-1)\text{.}$

Explanation.

To find $g(x-1)\text{,}$ we put in $(x-1)$ for the input. It is important to keep the parentheses because we have exponents and negative signs in the function.

\begin{align*} g(\substitute{x-1})\amp=2(\substitute{x-1})^2-3(\substitute{x-1})-5\\ \amp=2(x^2-2x+1)-3x+3-5\\ \amp=2x^2-4x+2-3x-2\\ \amp=2x^2-7x \end{align*}

Subsection 11.6.5 Technical Definition of a Function

In Section 5 we gave a formal definition of a function 11.5.2 and learned to identify what is and is not a function with sets or ordered pairs, tables and graphs. We also used the vertical line test 11.5.20.

Example 11.6.21. Formally Defining a Function.

We learned that sets of ordered pairs, tables and graphs can meet the formal definition of a function. Here is an example that shows a function in all three forms. We can verify that each input has at most one output.

$\{(1,4),(2,4),(3,3),(4,6),(5,-2)\}$

Figure 11.6.22. The function $f$ represented as a collection of ordered pairs.

$x$	$f(x)$
$1$	$4$
$2$	$4$
$3$	$3$
$4$	$6$
$5$	$-2$

Figure 11.6.23. The function $f$ represented as a table.

Figure 11.6.24. The function $f$ represented as a graph.

Example 11.6.25. Identifying What is Not a Function.

Identify whether each graph represents a function using the vertical line test 11.5.20.

Explanation.

Figure 11.6.29. A vertical line touching the graph twice makes this graph not give $y$ as a function of $x\text{.}$

Figure 11.6.30. A vertical line touching the graph twice makes this graph not give $y$ as a function of $x\text{.}$

Figure 11.6.31. All vertical lines only touch the graph once, so this graph does give $y$ as a function of $x\text{.}$

Exercises 11.6.6 Exercises

Function Basics.

1.

Teresa will spend ${\$135}$ to purchase some bowls and some plates. Each plate costs ${\$10}\text{,}$ and each bowl costs ${\$3}\text{.}$ The function $q(x)={-{\frac{10}{3}}x+45}$ models the number of bowls Teresa will purchase, where $x$ represents the number of plates to be purchased.

Interpret the meaning of $q(12)={5}\text{.}$

$12$ plates and $5$ bowls can be purchased.
$$12$ will be used to purchase bowls, and $$5$ will be used to purchase plates.
$$5$ will be used to purchase bowls, and $$12$ will be used to purchase plates.
$5$ plates and $12$ bowls can be purchased.

2.

Penelope will spend ${\$90}$ to purchase some bowls and some plates. Each plate costs ${\$2}\text{,}$ and each bowl costs ${\$3}\text{.}$ The function $q(x)={-{\frac{2}{3}}x+30}$ models the number of bowls Penelope will purchase, where $x$ represents the number of plates to be purchased.

Interpret the meaning of $q(21)={16}\text{.}$

$16$ plates and $21$ bowls can be purchased.
$$21$ will be used to purchase bowls, and $$16$ will be used to purchase plates.
$21$ plates and $16$ bowls can be purchased.
$$16$ will be used to purchase bowls, and $$21$ will be used to purchase plates.

Exercise Group.

3.

Evaluate the function at the given values.

$\displaystyle{g(x)={\frac{49}{x-3}}}$ .

$\displaystyle{g(10)=}$ .
$\displaystyle{g(3)=}$ .

4.

Evaluate the function at the given values.

$\displaystyle{h(x)={\frac{5}{x-8}}}$ .

$\displaystyle{h(3)=}$ .
$\displaystyle{h(8)=}$ .

5.

Use the graph of $F$ below to evaluate the given expressions. (Estimates are OK.)

$F(-8)={}$
$F(4)={}$

6.

Use the graph of $F$ below to evaluate the given expressions. (Estimates are OK.)

$F(-5)={}$
$F(5)={}$

7.

Use the table of values for $G$ below to evaluate the given expressions.

$x$	$-2$	$1$	$4$	$7$	$10$
$G(x)$	$1.2$	$7.3$	$0.5$	$0.8$	$2.6$

$G({4})={}$
$G({7})={}$

8.

Use the table of values for $H$ below to evaluate the given expressions.

$x$	$1$	$2$	$3$	$4$	$5$
$H(x)$	$9.9$	$7.2$	$6.5$	$1.9$	$7.5$

$H({1})={}$
$H({5})={}$

9.

Make a table of values for the function $g\text{,}$ defined by $g(x)={5x^{2}}\text{.}$ Based on values in the table, sketch a graph of $g\text{.}$

$x$	$g(x)$

$g(x)=5x^{2}$

10.

Make a table of values for the function $g\text{,}$ defined by $\displaystyle g(x)={\frac{2^{x}-5}{x^{2}+3}}\text{.}$ Based on values in the table, sketch a graph of $g\text{.}$

$x$	$g(x)$

$\displaystyle g(x)=\frac{2^{x}-5}{x^{2}+3}$

11.

The following figure has the graph $y=d(t)\text{,}$ which models a particle’s distance from the starting line in feet, where $t$ stands for time in seconds since timing started.

Find $d(9)\text{.}$
Interpret the meaning of $d(9)\text{.}$
1. The particle was $3$ feet away from the starting line $9$ seconds since timing started.
2. In the first $9$ seconds, the particle moved a total of $3$ feet.
3. In the first $3$ seconds, the particle moved a total of $9$ feet.
4. The particle was $9$ feet away from the starting line $3$ seconds since timing started.
Solve $d(t)={6}$ for $t\text{.}$ $t=$
Interpret the meaning of part c’s solution(s).
1. The particle was $6$ feet from the starting line $2$ seconds since timing started.
2. The particle was $6$ feet from the starting line $2$ seconds since timing started, and again $8$ seconds since timing started.
3. The particle was $6$ feet from the starting line $8$ seconds since timing started.
4. The particle was $6$ feet from the starting line $2$ seconds since timing started, or $8$ seconds since timing started.

12.

The following figure has the graph $y=d(t)\text{,}$ which models a particle’s distance from the starting line in feet, where $t$ stands for time in seconds since timing started.

Find $d(7)\text{.}$
Interpret the meaning of $d(7)\text{.}$
1. In the first $7$ seconds, the particle moved a total of $6$ feet.
2. The particle was $7$ feet away from the starting line $6$ seconds since timing started.
3. The particle was $6$ feet away from the starting line $7$ seconds since timing started.
4. In the first $6$ seconds, the particle moved a total of $7$ feet.
Solve $d(t)={2}$ for $t\text{.}$ $t=$
Interpret the meaning of part c’s solution(s).
1. The particle was $2$ feet from the starting line $1$ seconds since timing started, and again $9$ seconds since timing started.
2. The particle was $2$ feet from the starting line $9$ seconds since timing started.
3. The particle was $2$ feet from the starting line $1$ seconds since timing started, or $9$ seconds since timing started.
4. The particle was $2$ feet from the starting line $1$ seconds since timing started.

Domain and Range.

Find the domain and range of each function using its graph.

13.

A function is graphed.

This function has domain and range .

14.

A function is graphed.

This function has domain and range .

15.

A function is graphed.

The function has domain and range .

16.

A function is graphed.

The function has domain and range .

17.

A function is graphed.

The function has domain and range .

18.

A function is graphed.

The function has domain and range .

Exercise Group.

19.

Find the domain of $A$ where $\displaystyle{A(x)= \frac{\sqrt{20 + x}}{3 - x}}\text{.}$

20.

Find the domain of $p$ where $\displaystyle{p(x)= \frac{\sqrt{3 + x}}{10 - x}}\text{.}$

21.

An object was shot up into the air at an initial vertical speed of $384$ feet per second. Its height as time passes can be modeled by the quadratic function $f\text{,}$ where $f(t)={-16t^{2}+384t}\text{.}$ Here $t$ represents the number of seconds since the object’s release, and $f(t)$ represents the object’s height in feet.

Find the function’s domain and range in this context.

The function’s domain in this context is .

The function’s range in this context is .

22.

An object was shot up into the air at an initial vertical speed of $416$ feet per second. Its height as time passes can be modeled by the quadratic function $f\text{,}$ where $f(t)={-16t^{2}+416t}\text{.}$ Here $t$ represents the number of seconds since the object’s release, and $f(t)$ represents the object’s height in feet.

Find the function’s domain and range in this context.

The function’s domain in this context is .

The function’s range in this context is .

Using Technology to Explore Functions.

23.

Use technology to make a table of values for the function.

$F(x)={3x^{2}-13x+4}$

$x$	$F(x)$

24.

Use technology to make a table of values for the function.

$F(x)={0.15x^{2}+40x+11}$

$x$	$F(x)$

25.

Let $f(x)={-6970x-1116}\text{.}$ Choose an appropriate window for graphing $f$ that shows its key features.

The $x$-interval could be and the $y$-interval could be .

26.

Let $f(x)={776x-435}\text{.}$ Choose an appropriate window for graphing $f$ that shows its key features.

The $x$-interval could be and the $y$-interval could be .

27.

Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.

$y={\left(347+16x\right)\mathopen{}\left(223-x\right)}$ and $y={1000}$ intersect

?
zero times
one time
two times
three times

28.

Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.

$y={9x^{3}-3x^{2}-4x}$ and $y={-x+3}$ intersect

?
zero times
one time
two times
three times

29.

For the function $L$ defined by

\begin{equation*} L(x)=3000x^2+10x+4 \text{,} \end{equation*}

use technology to determine the following. Round answers as necessary.

Any intercepts.
The vertex.
The domain.
The range.

30.

For the function $M$ defined by

\begin{equation*} M(x)=-(300x-2950)^2 \text{,} \end{equation*}

use technology to determine the following. Round answers as necessary.

Any intercepts.
The vertex.
The domain.
The range.

31.

Let $f(x)=4x^2+5x-1$ and $g(x)=5\text{.}$ Use graphing technology to determine the following.

What are the points of intersection for these two functions?
Solve $f(x)=g(x)\text{.}$
Solve $f(x)\lt g(x)\text{.}$
Solve $f(x)\geq g(x)\text{.}$

32.

Let $p(x)=6x^2-3x+4$ and $k(x)=7\text{.}$ Use graphing technology to determine the following.

What are the points of intersection for these two functions?
Solve $p(x)=k(x)\text{.}$
Solve $p(x)\lt k(x)\text{.}$
Solve $p(x)\geq k(x)\text{.}$

33.

Use graphing technology to solve the equation $-0.02 x^2 + 1.97 x - 51.5=0.05\left(x-50\right)^2-0.03\left(x-50\right)\text{.}$ Approximate the solution(s) if necessary.

34.

Use graphing technology to solve the equation $-200x^2+60x-55=-20x-40\text{.}$ Approximate the solution(s) if necessary.

35.

Use graphing technology to solve the inequality $-15x^2-6\leq 10x-4\text{.}$ State the solution set using interval notation, and approximate if necessary.

36.

Use graphing technology to solve the inequality $\frac{1}{2}x^2+\frac{3}{2}x \geq \frac{1}{2}x-\frac{3}{2}\text{.}$ State the solution set using interval notation, and approximate if necessary.

Simplifying Expressions with Function Notation.

37.

Let $f$ be a function given by $f(x)={-5x^{2}-5x}\text{.}$ Find and simplify the following:

$f(x) - 2={}$
$f(x - 2)={}$
$-2f(x)={}$
$f(-2x)={}$

38.

Let $f$ be a function given by $f(x)={2x^{2}-x}\text{.}$ Find and simplify the following:

$f(x) - 3={}$
$f(x - 3)={}$
$-3f(x)={}$
$f(-3x)={}$

39.

Simplify $h({-x})\text{,}$ where $h(x)={-2+6x}\text{.}$

40.

Simplify $K(y)+3\text{,}$ where $K(y)={-3-6.1y}\text{.}$

41.

Simplify $7G(y)\text{,}$ where $G(y)={6y^{2}-3y-7}\text{.}$

42.

Simplify $g({-r})\text{,}$ where $g(r)={-7r^{2}+r-3}\text{.}$

Technical Definition of a Function.

43.

Does the following set of ordered pairs make for a function of $x\text{?}$

$\Big\{(4,10),(2,1),(-5,8),(3,8),(6,9)\Big\}$

This set of ordered pairs

?
describes
does not describe

a function of $x\text{.}$ This set of ordered pairs has domain and range .

44.

Does the following set of ordered pairs make for a function of $x\text{?}$

$\Big\{(3,5),(-1,10),(5,2),(5,0),(-9,7)\Big\}$

This set of ordered pairs

?
describes
does not describe

a function of $x\text{.}$ This set of ordered pairs has domain and range .

45.

Below is all of the information that exists about a function $K\text{.}$

$\begin{aligned} K(-1)\amp =4\amp K(1)\amp =0\amp K(3)\amp =-4 \end{aligned}$

Write $K$ as a set of ordered pairs.

$K$ has domain and range .

46.

Below is all of the information about a function $K\text{.}$

$\begin{aligned} K(a)\amp =5\amp K(b)\amp =7\\ K(c)\amp =0\amp K(d)\amp =5 \end{aligned}$

Write $K$ as a set of ordered pairs.

$K$ has domain and range .

47.

The following graphs show two relationships. Decide whether each graph shows a relationship where $y$ is a function of $x\text{.}$

The first graph

?
does
does not

give a function of $x\text{.}$ The second graph

?
does
does not

give a function of $x\text{.}$

48.

The following graphs show two relationships. Decide whether each graph shows a relationship where $y$ is a function of $x\text{.}$

The first graph

?
does
does not

give a function of $x\text{.}$ The second graph

?
does
does not

give a function of $x\text{.}$

49.

Some equations involving $x$ and $y$ define $y$ as a function of $x\text{,}$ and others do not. For example, if $x+y=1\text{,}$ we can solve for $y$ and obtain $y = 1-x\text{.}$ And we can then think of $y = f(x) = 1-x\text{.}$ On the other hand, if we have the equation $x=y^2$ then $y$ is not a function of $x\text{,}$ since for a given positive value of $x\text{,}$ the value of $y$ could equal $\sqrt{x}$ or it could equal $-\sqrt{x}\text{.}$ Select all of the following relations that make $y$ a function of $x\text{.}$ There are several correct answers.

$\displaystyle x+y=1$
$\displaystyle y^2 + x^2 = 1$
$\displaystyle \left|y\right| - x = 0$
$\displaystyle 4 x + 8 y + 2 = 0$
$\displaystyle y + x^2 = 1$
$\displaystyle y^3 + x^4 = 1$
$\displaystyle y^6 + x = 1$
$\displaystyle y - \left|x\right| = 0$

On the other hand, some equations involving $x$ and $y$ define $x$ as a function of $y$ (the other way round).

Select all of the following relations that make $x$ a function of $y\text{.}$ There are several correct answers.

$\displaystyle y^4 + x^5 = 1$
$\displaystyle \left|y\right| - x = 0$
$\displaystyle y^2 + x^2 = 1$
$\displaystyle 4 x + 8 y + 2 = 0$
$\displaystyle y - \left|x\right| = 0$

50.

$\displaystyle y^3 + x^4 = 1$
$\displaystyle y^2 + x^2 = 1$
$\displaystyle 6 x + 5 y + 5 = 0$
$\displaystyle y + x^2 = 1$
$\displaystyle y^6 + x = 1$
$\displaystyle y - \left|x\right| = 0$
$\displaystyle \left|y\right| - x = 0$
$\displaystyle x+y=1$

On the other hand, some equations involving $x$ and $y$ define $x$ as a function of $y$ (the other way round).

Select all of the following relations that make $x$ a function of $y\text{.}$ There are several correct answers.

$\displaystyle \left|y\right| - x = 0$
$\displaystyle 6 x + 5 y + 5 = 0$
$\displaystyle y - \left|x\right| = 0$
$\displaystyle y^2 + x^2 = 1$
$\displaystyle y^4 + x^5 = 1$

Exercise Group.

51.

Determine whether or not the following table could be the table of values of a function. If the table can not be the table of values of a function, give an input that has more than one possible output.

Input	Output
$2$	$2$
$4$	$-15$
$6$	$14$
$8$	$-18$
$-2$	$-12$

Could this be the table of values for a function?

If not, which input has more than one possible output?

-2
2
4
6
8
None, the table represents a function.

52.

Input	Output
$2$	$6$
$4$	$12$
$6$	$-9$
$8$	$12$
$-2$	$-13$

Could this be the table of values for a function?

If not, which input has more than one possible output?

-2
2
4
6
8
None, the table represents a function.

53.

Input	Output
$-4$	$-17$
$-3$	$-4$
$-2$	$-18$
$-3$	$26$
$-1$	$-9$

Could this be the table of values for a function?

If not, which input has more than one possible output?

-4
-3
-2
-1
None, the table represents a function.

54.

Input	Output
$-4$	$3$
$-3$	$-2$
$-2$	$-1$
$-3$	$12$
$-1$	$9$

Could this be the table of values for a function?

If not, which input has more than one possible output?

-4
-3
-2
-1
None, the table represents a function.

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input, \(x\)	output, \(\operatorname{f}(x)\)
\(-3\)	\(9\)
\(-2\)	\(4\)
\(-1\)	\(1\)
\(0\)	\(0\)
\(1\)	\(2\)
\(2\)	\(4\)
\(3\)	\(9\)

\(x\)	\(f(x)\)
\(-2\)	\(3(-2)+4=-2\)
\(-1\)	\(3(-1)+4=1\)
\(0 \)	\(3(0)+4=4\)
\(1 \)	\(3(1)+4=7\)
\(2 \)	\(3(2)+4=10\)

\(x\)	\(f(x)\)
\(1\)	\(4\)
\(2\)	\(4\)
\(3\)	\(3\)
\(4\)	\(6\)
\(5\)	\(-2\)

\(x\)	\(-2\)	\(1\)	\(4\)	\(7\)	\(10\)
\(G(x)\)	\(1.2\)	\(7.3\)	\(0.5\)	\(0.8\)	\(2.6\)

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(H(x)\)	\(9.9\)	\(7.2\)	\(6.5\)	\(1.9\)	\(7.5\)

Input	Output
\(-4\)	\(-17\)
\(-3\)	\(-4\)
\(-2\)	\(-18\)
\(-3\)	\(26\)
\(-1\)	\(-9\)

\(x\)	\(g(x)\)

\(x\)	\(g(x)\)

\(x\)	\(F(x)\)

\(x\)	\(F(x)\)