Stewart 9th Section 1.1: Four Ways to Represent a Function

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Stewart 9th Section 1.1: Four Ways to Represent a Function 0/72
1 Functions - Definition · Level 1
If \(f(x) = x + \sqrt{2 - x}\) and \(g(u) = u + \sqrt{2 - u}\), is it true that \(f = g\)?
2 Functions - Definition · Level 2
If \(f(x) = \dfrac{x^2 - x}{x - 1}\) and \(g(x) = x\), is it true that \(f = g\)?
3 Functions - Graph Reading · Level 2
The graph of a function \(g\) is given.
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(a) State the values of \(g(-2)\), \(g(0)\), \(g(2)\), and \(g(3)\).
(b) For what value(s) of \(x\) is \(g(x) = 3\)?
(c) For what value(s) of \(x\) is \(g(x) \leq 3\)?
(d) State the domain and range of \(g\).
(e) On what interval(s) is \(g\) increasing?

Enter your answer directly below each part above.

4 Functions - Graph Reading · Level 3
The graphs of \(f\) and \(g\) are given.
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(a) State the values of \(f(-4)\) and \(g(3)\).
(b) Which is larger, \(f(-3)\) or \(g(-3)\)?
(c) For what values of \(x\) is \(f(x) = g(x)\)?
(d) On what interval(s) is \(f(x) \leq g(x)\)?
(e) State the solution of the equation \(f(x) = -1\).
(f) On what interval(s) is \(g\) decreasing? (g) State the domain and range of \(f\). (h) State the domain and range of \(g\).

Enter your answer directly below each part above.

5 Functions - Applied · Level 2
Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.
6 Functions - Applied · Level 2
In this section we discussed examples of ordinary, everyday functions: population is a function of time, postage cost is a function of package weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.
7 Functions - Definition · Level 2
Determine whether the equation defines \(y\) as a function of \(x\). \(3 x - 5 y = 7\)
8 Functions - Definition · Level 2
Determine whether the equation defines \(y\) as a function of \(x\). \(3 x^2 - 2 y = 5\)
9 Functions - Definition · Level 2
Determine whether the equation defines \(y\) as a function of \(x\). \(x^2 + (y - 3)^2 = 5\)
10 Functions - Definition · Level 2
Determine whether the equation defines \(y\) as a function of \(x\). \(2 x y + 5 y^2 = 4\)
11 Functions - Definition · Level 2
Determine whether the equation defines \(y\) as a function of \(x\). \((y + 3)^3 + 1 = 2 x\)
12 Functions - Definition · Level 2
Determine whether the equation defines \(y\) as a function of \(x\). \(2 x - |y| = 0\)
13 Functions - Definition · Level 1
Determine whether the table defines \(y\) as a function of \(x\).
\(x\) (Height in inches) 72 60 60 63 70
\(y\) (Shoe size) 12 8 7 9 10
14 Functions - Definition · Level 1
Determine whether the table defines \(y\) as a function of \(x\).
\(x\) (Year) 2016 2017 2018 2019 2020
\(y\) (Tuition, \$) 10,900 11,000 11,200 11,200 11,300
15 Functions - Graph Reading · Level 2
Determine whether the curve is the graph of a function of \(x\). If it is, state the domain and range of the function.
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16 Functions - Graph Reading · Level 2
Determine whether the curve is the graph of a function of \(x\). If it is, state the domain and range of the function.
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17 Functions - Graph Reading · Level 2
Determine whether the curve is the graph of a function of \(x\). If it is, state the domain and range of the function.
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18 Functions - Graph Reading · Level 2
Determine whether the curve is the graph of a function of \(x\). If it is, state the domain and range of the function.
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19 Functions - Graph Reading · Level 2
The graph shows the global average temperature \(T\) during the 20th century.
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(a) What was the global average temperature in 1950?
(b) In what year was the average temperature 14.2 degrees C?
(c) In what year was the temperature smallest? Largest?
(d) What is the range of \(T\)?

Enter your answer directly below each part above.

20 Functions - Graph Reading · Level 2
The graph shows tree ring widths of Siberian pine trees from 1500 to 2000.
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(a) What is the range of the ring width function?
(b) What does the graph tend to say about the temperature of the earth? In particular, was there anything noteworthy about the volcanic eruptions in the mid-19th century?

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21 Functions - Sketch · Level 2
You put some ice cubes in a glass, fill the glass with cold water, and then let it sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.
22 Functions - Sketch · Level 2
You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.
23 Functions - Graph Reading · Level 2
The graph shows the power consumption \(P\) in megawatts in San Francisco for a day in September (\(t\) is measured in hours starting from midnight).
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(a) What was the power consumption at 6 AM? At 6 PM?
(b) When was the power consumption the lowest? When was it the highest?

Enter your answer directly below each part above.

24 Functions - Graph Reading · Level 2
Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?
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25 Functions - Sketch · Level 2
Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
26 Functions - Sketch · Level 2
Sketch a rough graph of the number of hours of daylight as a function of the time of year.
27 Functions - Sketch · Level 2
Sketch a rough graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.
28 Functions - Sketch · Level 2
Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.
29 Functions - Sketch · Level 2
A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.
30 Functions - Sketch · Level 3
An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If \(t\) represents the number of minutes since the plane has left the terminal, let \(x(t)\) be the horizontal distance traveled and \(y(t)\) be the altitude of the plane.
(a) Sketch a possible graph of \(x(t)\).
(b) Sketch a possible graph of \(y(t)\).
(c) Sketch a possible graph of the ground speed.
(d) Sketch a possible graph of the vertical velocity.

Enter your answer directly below each part above.

31 Functions - Graph Reading · Level 2
The temperature on a certain afternoon is recorded at half-hour intervals in the table below.
\(t\) (hours since noon) 0 2 4 6 8 10 12 14
\(T\) (degrees F) 74 69 68 66 70 78 82 86
(a) Sketch a rough graph of \(T\) as a function of \(t\).
(b) Use your graph to estimate the temperature at 9:00 AM.

Enter your answer directly below each part above.

32 Functions - Applied · Level 2
Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two alcoholic drinks). The table shows the data they obtained by averaging the BAC (in mg/mL) of the eight men.
\(t\) (hours) 0.2 0.5 0.75 1.0 1.25 1.5 2.0 2.5 3.0
BAC 0.25 0.41 0.40 0.33 0.29 0.24 0.18 0.12 0.07
(a) Use the data to sketch the graph of the BAC as a function of \(t\).
(b) Use your graph to describe how the effect of alcohol varies with time.

Enter your answer directly below each part above.

33 Functions - Evaluation · Level 2
If \(f(x) = 3 x^2 - x + 2\), find \(f(2)\), \(f(-2)\), \(f(a)\), \(f(-a)\), \(f(a + 1)\), \(2 f(a)\), \(f(2 a)\), \(f(a^2)\), \([f(a)]^2\), and \(f(a + h)\).
34 Functions - Evaluation · Level 2
If \(g(x) = \dfrac{x}{\sqrt{x + 1}}\), find \(g(0)\), \(g(3)\), \(5 g(a)\), \(\dfrac{1}{2} g(4 a)\), \(g(a^2)\), \([g(a)]^2\), \(g(a + h)\), and \(g(x - a)\).
35 Functions - Difference Quotient · Level 3
Evaluate the difference quotient for the given function. Simplify your answer. \(f(x) = 4 + 3 x - x^2\), \(\quad \dfrac{f(3 + h) - f(3)}{h}\)
36 Functions - Difference Quotient · Level 3
Evaluate the difference quotient for the given function. Simplify your answer. \(f(x) = x^3\), \(\quad \dfrac{f(a + h) - f(a)}{h}\)
37 Functions - Difference Quotient · Level 3
Evaluate the difference quotient for the given function. Simplify your answer. \(f(x) = \dfrac{1}{x}\), \(\quad \dfrac{f(x) - f(a)}{x - a}\)
38 Functions - Difference Quotient · Level 3
Evaluate the difference quotient for the given function. Simplify your answer. \(f(x) = \sqrt{x + 2}\), \(\quad \dfrac{f(x) - f(1)}{x - 1}\)
39 Functions - Domain · Level 2
Find the domain of the function. \(f(x) = \dfrac{x + 4}{x^2 - 9}\)
40 Functions - Domain · Level 2
Find the domain of the function. \(f(x) = \dfrac{x^2 + 1}{x^2 + 4 x - 21}\)
41 Functions - Domain · Level 2
Find the domain of the function. \(f(t) = \sqrt[3]{2 t - 1}\)
42 Functions - Domain · Level 3
Find the domain of the function. \(g(t) = \sqrt{3 - t} - \sqrt{2 + t}\)
43 Functions - Domain · Level 3
Find the domain of the function. \(h(x) = \dfrac{1}{\sqrt[4]{x^2 - 5 x}}\)
44 Functions - Domain · Level 3
Find the domain of the function. \(f(u) = \dfrac{u + 1}{1 + \dfrac{1}{u + 1}}\)
45 Functions - Domain · Level 3
Find the domain of the function. \(F(p) = \sqrt{2 - \sqrt{p}}\)
46 Functions - Domain · Level 3
Find the domain of the function. \(h(x) = \sqrt{x^2 - 4 x - 5}\)
47 Functions - Domain · Level 3
Find the domain and range of the function and sketch its graph. \(h(x) = \sqrt{4 - x^2}\)
48 Functions - Domain · Level 3
Find the domain of the function and sketch its graph. \(f(x) = \dfrac{x^2 - 4}{x - 2}\)
49 Functions - Piecewise · Level 2
Evaluate \(f(-3)\), \(f(0)\), and \(f(2)\) for the piecewise defined function. Then sketch the graph of the function. \( f(x) = \begin{cases} x^2 + 2 & \quad \text{if } x < 0 \\ x & \quad \text{if } x \geq 0 \end{cases} \)
50 Functions - Piecewise · Level 2
Evaluate \(f(-3)\), \(f(0)\), and \(f(2)\) for the piecewise defined function. Then sketch the graph of the function. \( f(x) = \begin{cases} 5 & \quad \text{if } x < 2 \\ \dfrac{1}{2} x - 3 & \quad \text{if } x \geq 2 \end{cases} \)
51 Functions - Piecewise · Level 2
Evaluate \(f(-3)\), \(f(0)\), and \(f(2)\) for the piecewise defined function. Then sketch the graph of the function. \( f(x) = \begin{cases} x + 1 & \quad \text{if } x \leq -1 \\ x^2 & \quad \text{if } x > -1 \end{cases} \)
52 Functions - Piecewise · Level 2
Evaluate \(f(-3)\), \(f(0)\), and \(f(2)\) for the piecewise defined function. Then sketch the graph of the function. \( f(x) = \begin{cases} -1 & \quad \text{if } x \leq 1 \\ 7 - 2 x & \quad \text{if } x > 1 \end{cases} \)
53 Functions - Absolute Value · Level 2
Sketch the graph of the function. \(f(x) = x + |x|\)
54 Functions - Absolute Value · Level 2
Sketch the graph of the function. \(f(x) = |x + 2|\)
55 Functions - Absolute Value · Level 2
Sketch the graph of the function. \(g(t) = |1 - 3 t|\)
56 Functions - Absolute Value · Level 2
Sketch the graph of the function. \(f(x) = \dfrac{|x|}{x}\)
57 Functions - Piecewise · Level 3
Sketch the graph of the function. \( f(x) = \begin{cases} |x| & \quad \text{if } |x| \leq 1 \\ 1 & \quad \text{if } |x| > 1 \end{cases} \)
58 Functions - Absolute Value · Level 3
Sketch the graph of the function. \(g(x) = ||x| - 1|\)
59 Functions - Formula from Graph · Level 3
Find a formula for the described function and sketch its graph. A line segment joining the points \((1, -3)\) and \((5, 7)\).
60 Functions - Formula from Graph · Level 3
Find a formula for the described function and sketch its graph. A line segment joining the points \((-5, 10)\) and \((7, -10)\).
61 Functions - Formula from Graph · Level 3
Find a formula for the described function and sketch its graph. The bottom half of the parabola \(x + (y - 1)^2 = 0\).
62 Functions - Formula from Graph · Level 3
Find a formula for the described function and sketch its graph. The top half of the circle \(x^2 + (y - 2)^2 = 4\).
63 Functions - Formula from Graph · Level 3
Find a formula for the function whose graph is given.
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64 Functions - Formula from Graph · Level 3
Find a formula for the function whose graph is given.
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65 Functions - Applied · Level 3
Find a formula for the described function and state its domain. A rectangle has a perimeter of 20 m. Express the area of the rectangle as a function of the length of one of its sides.
66 Functions - Applied · Level 3
Find a formula for the described function and state its domain. A rectangle has an area of 16 m\({}^2\). Express the perimeter of the rectangle as a function of the length of one of its sides.
67 Functions - Applied · Level 3
Find a formula for the described function and state its domain. Express the area of an equilateral triangle as a function of the length of a side.
68 Functions - Applied · Level 3
Find a formula for the described function and state its domain. A closed rectangular box with volume 8 ft\({}^3\) has its length twice its width. Express the height of the box as a function of the width.
69 Functions - Applied · Level 3
Find a formula for the described function and state its domain. An open rectangular box with volume 2 m\({}^3\) has a square base. Express the surface area of the box as a function of the length of a side of the base.
70 Functions - Applied · Level 3
Find a formula for the described function and state its domain. A right circular cylinder has volume 25 in\({}^3\). Express the radius of the cylinder as a function of the height.
71 Functions - Applied · Level 4
A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side \(x\) at each corner and then folding up the sides as in the figure. Express the volume \(V\) of the box as a function of \(x\).
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72 Functions - Applied · Level 4
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area \(A\) of the window as a function of the width \(x\) of the window.
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