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Section 11.6 Functions Chapter Review

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Subsection 11.6.1 Function Basics

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In Section 1 we defined functions informally, as well as function notation. We saw functions in four forms: verbal descriptions, formulas, graphs and tables.
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Example 11.6.1. Informal Definition of a Function.

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Determine whether each example below describes a function.
  1. The area of a circle given its radius.
  2. The number you square to get 9.
Explanation.
  1. 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)=πr2.
  2. 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. A function must give a single output for a given input.
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Example 11.6.2. Tables and Graphs.

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Make a table and a graph of the function f, where f(x)=x2.
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, 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)=x2
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Example 11.6.5. Translating between Four Descriptions of the Same Function.

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Consider a function f that triples its input and then adds 4. Translate this verbal description of f into a table, a graph, and a formula.
Explanation.
To make a table for f, 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, 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. 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, we have the formula
f(x)=3x+4.
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)
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Subsection 11.6.2 Domain and Range

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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.
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Example 11.6.8.

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Determine the domain of p, where p(x)=x2x−1.
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
2x−1=02x=1x=12
The domain is all real numbers except 12.
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Example 11.6.9.

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What is the domain of the function C, where C(x)=2x−3−5?
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
2x−3≥02x≥3x≥32
So on a number line, if we wanted to picture the domain of C, 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 [32,∞).
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Example 11.6.10. Range.

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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). 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, 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 (−∞,1].
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Subsection 11.6.3 Using Technology to Explore Functions

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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.
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Example 11.6.12. Finding an Appropriate Window.

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Graph the function t, where t(x)=(x+10)2−15, 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.
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Example 11.6.15. Using Technology to Determine Key Features of a Graph.

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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, where t(x)=(x+10)2−15.
Explanation.
From our graph we can now identify the vertex at (−10,−15), the y-intercept at (0,85), and the x-intercepts at approximately (−13.9,0) and (−6.13,0).
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.
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Example 11.6.17. Solving Equations Graphically Using Technology.

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Use graphing technology to solve the equation t(x)=40, where t(x)=(x+10)2−15.
Explanation.
To solve the equation t(x)=40, 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). The solution set is {−17.4,−2.58}.
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.
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Subsection 11.6.4 Simplifying Expressions with Function Notation

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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 13f(x). Here are some examples.
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Example 11.6.19. Negative Signs in and out of Function Notation.

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Find and simplify a formula for f(−x) and −f(x), where f(x)=−3x2−7x+1.
Explanation.
To find f(−x), we use an input of −x in our function f and simplify to get:
f(−x)=−3(−x)2−7(−x)+1=−3x2+7x+1
To find −f(x), we take the opposite of the function f and simplify to get:
−f(x)=−(−3x2−7x+1)=3x2+7x−1
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Example 11.6.20. Other Nontrivial Simplifications.

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If g(x)=2x2−3x−5, find and simplify a formula for g(x−1).
Explanation.
To find g(x−1), 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.
g(x−1)=2(x−1)2−3(x−1)−5=2(x2−2x+1)−3x+3−5=2x2−4x+2−3x−2=2x2−7x
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Subsection 11.6.5 Technical Definition of a Function

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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.
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Example 11.6.21. Formally Defining a Function.

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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.
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{(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.
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Example 11.6.25. Identifying What is Not a Function.

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Identify whether each graph represents a function using the vertical line test 11.5.20.
Figure 11.6.26.
Figure 11.6.27.
Figure 11.6.28.
Explanation.
Figure 11.6.29. A vertical line touching the graph twice makes this graph not give y as a function of x.
Figure 11.6.30. A vertical line touching the graph twice makes this graph not give y as a function of x.
Figure 11.6.31. All vertical lines only touch the graph once, so this graph does give y as a function of x.
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Exercises 11.6.6 Exercises

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Function Basics.

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1.
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Teresa will spend $135 to purchase some bowls and some plates. Each plate costs $10, and each bowl costs $3. The function q(x)=−103x+45 models the number of bowls Teresa will purchase, where x represents the number of plates to be purchased.
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Interpret the meaning of q(12)=5.
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  1. 12 plates and 5 bowls can be purchased.
  2. $12 will be used to purchase bowls, and $5 will be used to purchase plates.
  3. $5 will be used to purchase bowls, and $12 will be used to purchase plates.
  4. 5 plates and 12 bowls can be purchased.
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2.
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Penelope will spend $90 to purchase some bowls and some plates. Each plate costs $2, and each bowl costs $3. The function q(x)=−23x+30 models the number of bowls Penelope will purchase, where x represents the number of plates to be purchased.
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Interpret the meaning of q(21)=16.
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  1. 16 plates and 21 bowls can be purchased.
  2. $21 will be used to purchase bowls, and $16 will be used to purchase plates.
  3. 21 plates and 16 bowls can be purchased.
  4. $16 will be used to purchase bowls, and $21 will be used to purchase plates.
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Exercise Group.

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3.
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Evaluate the function at the given values.
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g(x)=49x−3 .
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  1. g(10)= .
  2. g(3)= .
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4.
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Evaluate the function at the given values.
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h(x)=5x−8 .
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  1. h(3)= .
  2. h(8)= .
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5.
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Use the graph of F below to evaluate the given expressions. (Estimates are OK.)
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  1. F(−8)=
  2. F(4)=
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6.
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Use the graph of F below to evaluate the given expressions. (Estimates are OK.)
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  1. F(−5)=
  2. F(5)=
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7.
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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
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  1. G(4)=
  2. G(7)=
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8.
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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
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  1. H(1)=
  2. H(5)=
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9.
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Make a table of values for the function g, defined by g(x)=5x2. Based on values in the table, sketch a graph of g.
x g(x)
g(x)=5x2
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10.
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Make a table of values for the function g, defined by g(x)=2x−5x2+3. Based on values in the table, sketch a graph of g.
x g(x)
g(x)=2x−5x2+3
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11.

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The following figure has the graph y=d(t), which models a particle’s distance from the starting line in feet, where t stands for time in seconds since timing started.
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  1. Find d(9).
  2. Interpret the meaning of d(9).
    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.
  3. Solve d(t)=6 for t. t=
  4. 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.
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12.

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The following figure has the graph y=d(t), which models a particle’s distance from the starting line in feet, where t stands for time in seconds since timing started.
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  1. Find d(7).
  2. Interpret the meaning of d(7).
    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.
  3. Solve d(t)=2 for t. t=
  4. 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.
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Domain and Range.

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Find the domain and range of each function using its graph.
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13.
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A function is graphed.
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This function has domain and range .
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14.
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A function is graphed.
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This function has domain and range .
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15.
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A function is graphed.
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The function has domain and range .
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16.
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A function is graphed.
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The function has domain and range .
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17.
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A function is graphed.
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The function has domain and range .
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18.
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A function is graphed.
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The function has domain and range .
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Exercise Group.

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19.
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Find the domain of A where A(x)=20+x3−x.
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20.
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Find the domain of p where p(x)=3+x10−x.
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21.

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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, where f(t)=−16t2+384t. Here t represents the number of seconds since the object’s release, and f(t) represents the object’s height in feet.
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Find the function’s domain and range in this context.
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The function’s domain in this context is .
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The function’s range in this context is .
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22.

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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, where f(t)=−16t2+416t. Here t represents the number of seconds since the object’s release, and f(t) represents the object’s height in feet.
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Find the function’s domain and range in this context.
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The function’s domain in this context is .
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The function’s range in this context is .
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Using Technology to Explore Functions.

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23.
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Use technology to make a table of values for the function.
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F(x)=3x2−13x+4
x F(x)
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24.
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Use technology to make a table of values for the function.
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F(x)=0.15x2+40x+11
x F(x)
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25.
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Let f(x)=−6970x−1116. Choose an appropriate window for graphing f that shows its key features.
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The x-interval could be and the y-interval could be .
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26.
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Let f(x)=776x−435. Choose an appropriate window for graphing f that shows its key features.
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The x-interval could be and the y-interval could be .
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27.
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Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
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y=(347+16x)(223−x) and y=1000 intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
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.
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28.
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Use technology to make some graphs and determine how many times the graphs of the following curves cross each other.
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y=9x3−3x2−4x and y=−x+3 intersect
  • ?
  • zero times
  • one time
  • two times
  • three times
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.
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29.
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For the function L defined by
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L(x)=3000x2+10x+4,
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use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.
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30.
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For the function M defined by
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M(x)=−(300x−2950)2,
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use technology to determine the following. Round answers as necessary.
  1. Any intercepts.
  2. The vertex.
  3. The domain.
  4. The range.
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31.
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Let f(x)=4x2+5x−1 and g(x)=5. Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve f(x)=g(x).
  3. Solve f(x)<g(x).
  4. Solve f(x)≥g(x).
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32.
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Let p(x)=6x2−3x+4 and k(x)=7. Use graphing technology to determine the following.
  1. What are the points of intersection for these two functions?
  2. Solve p(x)=k(x).
  3. Solve p(x)<k(x).
  4. Solve p(x)≥k(x).
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33.
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Use graphing technology to solve the equation −0.02x2+1.97x−51.5=0.05(x−50)2−0.03(x−50). Approximate the solution(s) if necessary.
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34.
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Use graphing technology to solve the equation −200x2+60x−55=−20x−40. Approximate the solution(s) if necessary.
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35.
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Use graphing technology to solve the inequality −15x2−6≤10x−4. State the solution set using interval notation, and approximate if necessary.
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36.
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Use graphing technology to solve the inequality 12x2+32x≥12x−32. State the solution set using interval notation, and approximate if necessary.
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Simplifying Expressions with Function Notation.

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37.
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Let f be a function given by f(x)=−5x2−5x. Find and simplify the following:
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  1. f(x)−2=
  2. f(x−2)=
  3. −2f(x)=
  4. f(−2x)=
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38.
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Let f be a function given by f(x)=2x2−x. Find and simplify the following:
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  1. f(x)−3=
  2. f(x−3)=
  3. −3f(x)=
  4. f(−3x)=
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39.
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Simplify h(−x), where h(x)=−2+6x.
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40.
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Simplify K(y)+3, where K(y)=−3−6.1y.
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41.
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Simplify 7G(y), where G(y)=6y2−3y−7.
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42.
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Simplify g(−r), where g(r)=−7r2+r−3.
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Technical Definition of a Function.

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43.
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Does the following set of ordered pairs make for a function of x?
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{(4,10),(2,1),(−5,8),(3,8),(6,9)}
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This set of ordered pairs
  • ?
  • describes
  • does not describe
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a function of x. This set of ordered pairs has domain and range .
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44.
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Does the following set of ordered pairs make for a function of x?
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{(3,5),(−1,10),(5,2),(5,0),(−9,7)}
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This set of ordered pairs
  • ?
  • describes
  • does not describe
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a function of x. This set of ordered pairs has domain and range .
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45.
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Below is all of the information that exists about a function K.
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K(−1)=4K(1)=0K(3)=−4
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Write K as a set of ordered pairs.
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K has domain and range .
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46.
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Below is all of the information about a function K.
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K(a)=5K(b)=7K(c)=0K(d)=5
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Write K as a set of ordered pairs.
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K has domain and range .
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47.
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The following graphs show two relationships. Decide whether each graph shows a relationship where y is a function of x.
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The first graph
  • ?
  • does
  • does not
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give a function of x. The second graph
  • ?
  • does
  • does not
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give a function of x.
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48.
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The following graphs show two relationships. Decide whether each graph shows a relationship where y is a function of x.
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🔗
The first graph
  • ?
  • does
  • does not
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give a function of x. The second graph
  • ?
  • does
  • does not
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give a function of x.
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49.
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Some equations involving x and y define y as a function of x, and others do not. For example, if x+y=1, we can solve for y and obtain y=1−x. And we can then think of y=f(x)=1−x. On the other hand, if we have the equation x=y2 then y is not a function of x, since for a given positive value of x, the value of y could equal x or it could equal −x. Select all of the following relations that make y a function of x. There are several correct answers.
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  • x+y=1
  • y2+x2=1
  • |y|−x=0
  • 4x+8y+2=0
  • y+x2=1
  • y3+x4=1
  • y6+x=1
  • y−|x|=0
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On the other hand, some equations involving x and y define x as a function of y (the other way round).
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Select all of the following relations that make x a function of y. There are several correct answers.
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  • y4+x5=1
  • |y|−x=0
  • y2+x2=1
  • 4x+8y+2=0
  • y−|x|=0
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50.
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Some equations involving x and y define y as a function of x, and others do not. For example, if x+y=1, we can solve for y and obtain y=1−x. And we can then think of y=f(x)=1−x. On the other hand, if we have the equation x=y2 then y is not a function of x, since for a given positive value of x, the value of y could equal x or it could equal −x. Select all of the following relations that make y a function of x. There are several correct answers.
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  • y3+x4=1
  • y2+x2=1
  • 6x+5y+5=0
  • y+x2=1
  • y6+x=1
  • y−|x|=0
  • |y|−x=0
  • x+y=1
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On the other hand, some equations involving x and y define x as a function of y (the other way round).
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Select all of the following relations that make x a function of y. There are several correct answers.
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  • |y|−x=0
  • 6x+5y+5=0
  • y−|x|=0
  • y2+x2=1
  • y4+x5=1
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Exercise Group.

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51.
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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
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🔗
Could this be the table of values for a function?
  • yes
  • no
🔗
🔗
If not, which input has more than one possible output?
  • -2
  • 2
  • 4
  • 6
  • 8
  • None, the table represents a function.
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52.
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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 6
4 12
6 −9
8 12
−2 −13
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🔗
Could this be the table of values for a function?
  • yes
  • no
🔗
🔗
If not, which input has more than one possible output?
  • -2
  • 2
  • 4
  • 6
  • 8
  • None, the table represents a function.
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53.
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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
−4 −17
−3 −4
−2 −18
−3 26
−1 −9
🔗
🔗
Could this be the table of values for a function?
  • yes
  • no
🔗
🔗
If not, which input has more than one possible output?
  • -4
  • -3
  • -2
  • -1
  • None, the table represents a function.
🔗
54.
🔗
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
−4 3
−3 −2
−2 −1
−3 12
−1 9
🔗
🔗
Could this be the table of values for a function?
  • yes
  • no
🔗
🔗
If not, which input has more than one possible output?
  • -4
  • -3
  • -2
  • -1
  • None, the table represents a function.
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