Calculus Single Variable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 1

Foundation for Calculus: Functions and Limits - all with Video Answers

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Section 1

Functions and Change

Problem 1

The population of a city, $P,$ in millions, is a function of $t,$ the number of years since $2010,$ so $P=f(t) .$ Explain the meaning of the statement $f(5)=7$ in terms of the population of this city.

Gregory H.

Gregory H.

Numerade Educator

Problem 2

The pollutant PCB (polychlorinated biphenyl) can affect the thickness of pelican eggshells. Thinking of the thickness, $T,$ of the eggshells, in $\mathrm{mm},$ as a function of the concentration, $P,$ of $\mathrm{PCBs}$ in ppm (parts per million), we have $T=f(P)$. Explain the meaning of $f(200)$ in terms of thickness of pelican eggs and concentration of $\mathrm{PCBs}$

Gregory H.

Numerade Educator

Problem 3

Describe what Figure 1.8 tells you about an assembly line whose productivity is represented as a function of the number of workers on the line.

Gregory H.

Numerade Educator

Problem 4

Find an equation for the line that passes through the given points.
(0,0) and (1,1)

Natalie A.

Numerade Educator

Problem 5

Find an equation for the line that passes through the given points.
(0,2) and (2,3)

Natalie A.

Numerade Educator

Problem 6

Find an equation for the line that passes through the given points.
(-2,1) and (2,3)

Natalie A.

Numerade Educator

Problem 7

Find an equation for the line that passes through the given points.
(-1,0) and (2,6)

Yujian Z.

Numerade Educator

Problem 8

Determine the slope and the $y$-intercept of the line whose equation is given
$$2 y+5 x-8=0$$

Gregory H.

Numerade Educator

Problem 9

Determine the slope and the $y$-intercept of the line whose equation is given
$$7 y+12 x-2=0$$

Natalie A.

Numerade Educator

Problem 10

Determine the slope and the $y$-intercept of the line whose equation is given
$$-4 y+2 x+8=0$$

Natalie A.

Numerade Educator

Problem 11

Determine the slope and the $y$-intercept of the line whose equation is given
$$12 x=6 y+4$$

Natalie A.

Numerade Educator

Problem 12

Match the graphs in Figure 1.9 with the following equations. (Note that the $x$ and $y$ scales may be unequal.)
(a) $y=x-5$
(b) $-3 x+4=y$
(c) $\quad 5=y$
(d) $y=-4 x-5$
(e) $\quad y=x+6$
(f) $y=x / 2$

Gregory H.

Numerade Educator

Problem 13

Match the graphs in Figure 1.10 with the following equations. (Note that the $x$ and $y$ scales may be unequal.)
(a) $y=-2.72 x$
(b) $y=0.01+0.001 x$
(c) $y=27.9-0.1 x$
(d) $y=0.1 x-27.9$
(e) $\quad y=-5.7-200 x$
(f) $y=x / 3.14$

Gregory H.

Numerade Educator

Problem 14

Estimate the slope and the equation of the line in Figure 1.11

Gregory H.

Numerade Educator

Problem 15

Find an equation for the line with slope $m$ through the point $(a, c)$

Gregory H.

Numerade Educator

Problem 16

Find a linear function that generates the values in Table $1.3 .$
$$\begin{array}{c|r|r|r|r|r}
\hline x & 5.2 & 5.3 & 5.4 & 5.5 & 5.6 \\
\hline y & 27.8 & 29.2 & 30.6 & 32.0 & 33.4 \\
\hline
\end{array}$$

Lucía G.

Numerade Educator

Problem 17

Use the facts that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals of one another.
Find an equation for the line through the point (2,1) which is perpendicular to the line $y=5 x-3$.

Gregory H.

Numerade Educator

Problem 18

Find an equation for the line through the point (2,1) which is perpendicular to the line $y=5 x-3$.
Find equations for the lines through the point (1,5) that are parallel to and perpendicular to the line with equation $y+4 x=7$

Gregory H.

Numerade Educator

Problem 19

Use the facts that parallel lines have equal slopes and that the slopes of perpendicular lines are negative reciprocals of one another.
Find equations for the lines through the point $(a, b)$ that are parallel and perpendicular to the line $y=m x+c$ assuming $m \neq 0$

Gregory H.

Numerade Educator

Problem 20

Give the approximate domain and range of each function. Assume the entire graph is shown

Gregory H.

Numerade Educator

Problem 21

Give the approximate domain and range of each function. Assume the entire graph is shown

Gregory H.

Numerade Educator

Problem 22

Give the approximate domain and range of each function. Assume the entire graph is shown

Gregory H.

Numerade Educator

Problem 23

Give the approximate domain and range of each function. Assume the entire graph is shown

Gregory H.

Numerade Educator

Problem 24

Find the domain and range
$$y=x^{2}+2$$

Gregory H.

Numerade Educator

Problem 25

Find the domain and range
$$y=\frac{1}{x^{2}+2}$$

Gregory H.

Numerade Educator

Problem 26

If $f(t)=\sqrt{t^{2}-16},$ find all values of $t$ for which $f(t)$ is a real number. Solve $f(t)=3$

Gregory H.

Numerade Educator

Problem 27

Write a formula representing the function.
The volume of a sphere is proportional to the cube of its radius, $r$.

Gregory H.

Numerade Educator

Problem 28

Write a formula representing the function.
The average velocity, $v,$ for a trip over a fixed distance, $d,$ is inversely proportional to the time of travel, $t$

Patrick V.

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Problem 29

Write a formula representing the function.
The strength, $S$, of a beam is proportional to the square of its thickness, $h$

Patrick V.

Numerade Educator

Problem 30

Write a formula representing the function.
The energy, $E$, expended by a swimming dolphin is proportional to the cube of the speed, $v$, of the dolphin.

Patrick V.

Numerade Educator

Problem 31

Write a formula representing the function.
The number of animal species, $N,$ of a certain body length, $l,$ is inversely proportional to the square of $l$

Gregory H.

Numerade Educator

Problem 32

In December 2010 , the snowfall in Minneapolis was unusually high, ${ }^{2}$ leading to the collapse of the roof of the Metrodome. Figure 1.12 gives the snowfall, $S$, in Minneapolis for December $6-15,2010$.
(a) How do you know that the snowfall data represents a function of date?
(b) Estimate the snowfall on December 12 .
(c) On which day was the snowfall more than 10 inches?
(d) During which consecutive two-day interval was the increase in snowfall largest?

Robert D.

Numerade Educator

Problem 33

The value of a car, $V=f(a),$ in thousands of dollars, is a function of the age of the car, $a$, in years.
(a) Interpret the statement $f(5)=6$
(b) Sketch a possible graph of $V$ against $a$. Is $f$ an increasing or decreasing function? Explain.
(c) Explain the significance of the horizontal and vertical intercepts in terms of the value of the car.

Gregory H.

Numerade Educator

Problem 34

Which graph in Figure 1.13 best matches each of the following stories? $^{3}$ Write a story for the remaining graph.
(a) I had just left home when I realized I had forgotten my books, so I went back to pick them up.
(b) Things went fine until I had a flat tire.
(c) I started out calmly but sped up when I realized I was going to be late.

Gregory H.

Numerade Educator

Problem 35

The function $S=f(t)$ gives the average annual sea level, $S,$ in meters, in Aberdeen, Scotland, $^{4}$ as a function of $t,$ the number of years before 2012 . Write a mathematical expression that represents the given statement.
In 2000 the average annual sea level in Aberdeen was 7.049 meters.

Gregory H.

Numerade Educator

Problem 36

The function $S=f(t)$ gives the average annual sea level, $S,$ in meters, in Aberdeen, Scotland, $^{4}$ as a function of $t,$ the number of years before 2012 . Write a mathematical expression that represents the given statement.
The average annual sea level in Aberdeen in 2012 .

Gregory H.

Numerade Educator

Problem 37

The function $S=f(t)$ gives the average annual sea level, $S,$ in meters, in Aberdeen, Scotland, $^{4}$ as a function of $t,$ the number of years before 2012 . Write a mathematical expression that represents the given statement.
The average annual sea level in Aberdeen was the same in 1949 and 2000 .

Gregory H.

Numerade Educator

Problem 38

The function $S=f(t)$ gives the average annual sea level, $S,$ in meters, in Aberdeen, Scotland, $^{4}$ as a function of $t,$ the number of years before 2012 . Write a mathematical expression that represents the given statement.
The average annual sea level in Aberdeen decreased by 8 millimeters from 2011 to 2012 .

Gregory H.

Numerade Educator

Problem 39

Problems $39-42$ ask you to plot graphs based on the following story: "As I drove down the highway this morning, at first traffic was fast and uncongested, then it crept nearly bumperto-bumper until we passed an accident, after which traffic flow went back to normal until I exited."
Driving speed against time on the highway

Gregory H.

Numerade Educator

Problem 40

Ask you to plot graphs based on the following story: "As I drove down the highway this morning, at first traffic was fast and uncongested, then it crept nearly bumperto-bumper until we passed an accident, after which traffic flow went back to normal until I exited."
Distance driven against time on the highway

Gregory H.

Numerade Educator

Problem 41

Ask you to plot graphs based on the following story: "As I drove down the highway this morning, at first traffic was fast and uncongested, then it crept nearly bumperto-bumper until we passed an accident, after which traffic flow went back to normal until I exited."
Distance from my exit vs time on the highway

Kian M.

Numerade Educator

Problem 42

Ask you to plot graphs based on the following story: "As I drove down the highway this morning, at first traffic was fast and uncongested, then it crept nearly bumperto-bumper until we passed an accident, after which traffic flow went back to normal until I exited."
Distance between cars vs distance driven on the high-way

Kian M.

Numerade Educator

Problem 43

An object is put outside on a cold day at time $t=0 .$ Its temperature, $H=f(t),$ in ${ }^{\circ} \mathrm{C},$ is graphed in Figure 1.14
(a) What does the statement $f(30)=10$ mean in terms of temperature? Include units for 30 and for 10 in your answer.
(b) Explain what the vertical intercept, $a$, and the horizontal intercept, $b$, represent in terms of temperature of the object and time outside.

Gregory H.

Numerade Educator

Problem 44

A rock is dropped from a window and falls to the ground below. The height, $s$ (in meters), of the rock above ground is a function of the time, $t$ (in seconds), since the rock was dropped, so $s=f(t)$.
(a) Sketch a possible graph of $s$ as a function of $t$
(b) Explain what the statement $f(7)=12$ tells us about the rock's fall.
(c) The graph drawn as the answer for part (a) should have a horizontal and vertical intercept. Interpret each intercept in terms of the rock's fall.

Gregory H.

Numerade Educator

Problem 45

You drive at a constant speed from Chicago to Detroit, a distance of 275 miles. About 120 miles from Chicago you pass through Kalamazoo, Michigan. Sketch a graph of your distance from Kalamazoo as a function of time.

Gregory H.

Numerade Educator

Problem 46

US imports of crude oil and petroleum have been increasing. $^{5}$ There have been many ups and downs, but the general trend is shown by the line in Figure $1.15 .$
(a) Find the slope of the line. Include its units of measurement.
(b) Write an equation for the line. Define your variables, including their units.
(c) Assuming the trend continues, when does the linear model predict imports will reach 18 million barrels per day? Do you think this is a reliable prediction? Give reasons.

Kian M.

Numerade Educator

Problem 47

Use Figure 1.16 showing how the quantity, $Q,$ of grass (kg/hectare) in different parts of Namibia depended on the average annual rainfall, $r,(\mathrm{mm}),$ in two different years.
(a) For $1939,$ find the slope of the line, including units.
(b) Interpret the slope in this context.
(c) Find the equation of the line.

Gregory H.

Numerade Educator

Problem 48

Use Figure 1.16 showing how the quantity, $Q,$ of grass (kg/hectare) in different parts of Namibia depended on the average annual rainfall, $r,(\mathrm{mm}),$ in two different years.
(a) For $1997,$ find the slope of the line, including units.
(b) Interpret the slope in this context.
(c) Find the equation of the line.

Kian M.

Numerade Educator

Problem 49

Use Figure 1.16 showing how the quantity, $Q,$ of grass (kg/hectare) in different parts of Namibia depended on the average annual rainfall, $r,(\mathrm{mm}),$ in two different years.
Which of the two functions in Figure 1.16 has the larger difference quotient $\Delta Q / \Delta r ?$ What does this tell us about grass in Namibia?

Kian M.

Numerade Educator

Problem 50

Marmots are large squirrels that hibernate in the winter and come out in the spring. Figure 1.17 shows the date (days after Jan 1) that they are first sighted each year in Colorado as a function of the average minimum daily temperature for that year.
(a) Find the slope of the line, including units.
(b) What does the sign of the slope tell you about marmots?
(c) Use the slope to determine how much difference $6^{\circ} \mathrm{C}$ warming makes to the date of first appearance of a marmot.
(d) Find the equation of the line.

Kian M.

Numerade Educator

Problem 51

In Colorado spring has arrived when the bluebell first flowers. Figure 1.18 shows the date (days after Jan 1 ) that the first flower is sighted in one location as a function of the first date (days after Jan 1 ) of bare (snowfree) ground. $^{8}$
(a) If the first date of bare ground is $140,$ how many days later is the first bluebell flower sighted?
(b) Find the slope of the line, including units.
(c) What does the sign of the slope tell you about bluebells?
(d) Find the equation of the line.

Kian M.

Numerade Educator

Problem 52

On March $5,2015,$ Capracotta, Italy, received $256 \mathrm{cm}$ $(100.787 \text { inches) of snow in } 18$ hours.
(a) Assuming the snow fell at a constant rate and there were already $100 \mathrm{cm}$ of snow on the ground, find a formula for $f(t),$ in $\mathrm{cm},$ for the depth of snow as a function of $t$ hours since the snowfall began on March 5
(b) What are the domain and range of $f ?$

Gregory H.

Numerade Educator

Problem 53

In a California town, the monthly charge for waste collection is $\$ 8$ for 32 gallons of waste and $\$ 12.32$ for 68 gallons of waste.
(a) Find a linear formula for the cost, $C,$ of waste collection as a function of the number of gallons of waste, $w$
(b) What is the slope of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection.
(c) What is the vertical intercept of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection.

Natalie A.

Numerade Educator

Problem 54

For tax purposes, you may have to report the value of your assets, such as cars or refrigerators. The value you report drops with time. "Straight-line depreciation" assumes that the value is a linear function of time. If a \$950 refrigerator depreciates completely in seven years, find a formula for its value as a function of time.

Gregory H.

Numerade Educator

Problem 55

Residents of the town of Maple Grove who are connected to the municipal water supply are billed a fixed amount monthly plus a charge for each cubic foot of water used. A household using 1000 cubic feet was billed $\$ 40,$ while one using 1600 cubic feet was billed $\$ 55$
(a) What is the charge per cubic foot?
(b) Write an equation for the total cost of a resident's water as a function of cubic feet of water used.
(c) How many cubic feet of water used would lead to a bill of $\$ 100 ?$

Gregory H.

Numerade Educator

Problem 56

A controversial 1992 Danish study $^{10}$ reported that men's average sperm count decreased from 113 million per milliliter in 1940 to 66 million per milliliter in 1990 .
(a) Express the average sperm count, $S,$ as a linear function of the number of years, $t,$ since 1940 .
(b) A man's fertility is affected if his sperm count drops below about 20 million per milliliter. If the linear model found in part (a) is accurate, in what year will the average male sperm count fall below this level?

Gregory H.

Numerade Educator

Problem 57

Let $f(t)$ be the number of US billionaires in year $t$
(a) Express the following statements $^{11}$ in terms of $f$
(i) In 2001 there were 272 US billionaires.
(ii) In 2014 there were 525 US billionaires.
(b) Find the average yearly increase in the number of US billionaires from 2001 to 2014 . Express this using $f$
(c) Assuming the yearly increase remains constant, find a formula predicting the number of US billionaires in year $t$

Gregory H.

Numerade Educator

Problem 58

The cost of planting seed is usually a function of the number of acres sown. The cost of the equipment is a fixed cost because it must be paid regardless of the number of acres planted. The costs of supplies and labor vary with the number of acres planted and are called variable costs. Suppose the fixed costs are $\$ 10,000$ and the variable costs are $\$ 200$ per acre. Let $C$ be the total cost, measured in thousands of dollars, and let $x$ be the number of acres planted.
(a) Find a formula for $C$ as a function of $x$.
(b) Graph $C$ against $x$.
(c) Which feature of the graph represents the fixed costs? Which represents the variable costs?

Gregory H.

Numerade Educator

Problem 59

An airplane uses a fixed amount of fuel for takeoff, a (different) fixed amount for landing, and a third fixed amount per mile when it is in the air. How does the total quantity of fuel required depend on the length of the trip? Write a formula for the function involved. Explain the meaning of the constants in your formula.

Kian M.

Numerade Educator

Problem 60

For the line $y=f(x)$ in Figure $1.19,$ evaluate
(a) $f(423)-f(422)$
(b) $f(517)-f(513)$

Gregory H.

Numerade Educator

Problem 61

For the line $y=g(x)$ in Figure $1.20,$ evaluate
(a) $g(4210)-g(4209)$
(b) $g(3760)-g(3740)$

Gregory H.

Numerade Educator

Problem 62

An alternative to petroleum-based diesel fuel, biodiesel, is derived from renewable resources such as food crops, algae, and animal oils. The table shows the recent annual percent growth in US biodiesel exports. ${ }^{12}$
(a) Find the largest time interval over which the percentage growth in the US exports of biodiesel was an increasing function of time. Interpret what increasing means, practically speaking, in this case.
(b) Find the largest time interval over which the actual US exports of biodiesel was an increasing function of time. Interpret what increasing means, practically speaking, in this case.
$$\begin{array}{c|c|c|c|c|c}
\hline \text { Year } & 2010 & 2011 & 2012 & 2013 & 2014 \\
\hline \text { \% growth over previous yr } & -60.5 & -30.5 & 69.9 & 53.0 & -57.8 \\
\hline
\end{array}$$

Kian M.

Numerade Educator

Problem 63

Hydroelectric power is electric power generated by the force of moving water. Figure 1.21 shows ${ }^{13}$ the annual percent growth in hydroelectric power consumption by the US industrial sector between 2006 and 2014 .
(a) Find the largest time interval over which the percentage growth in the US consumption of hydroelectric power was an increasing function of time. Interpret what increasing means, practically speaking, in this case.
(b) Find the largest time interval over which the actual US consumption of hydroelectric power was a decreasing function of time. Interpret what decreasing means, practically speaking, in this case.

Kian M.

Numerade Educator

Problem 64

Solar panels are arrays of photovoltaic cells that convert solar radiation into electricity. The table shows the annual percent change in the US price per watt of a solar panel. $^{14}$
$$\begin{array}{c|c|c|c|c|c|c}
\hline \text { Year } & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 \\
\hline \text { \% growth over previous yr } & 6.7 & 9.7 & -3.7 & 3.6 & -20.1 & -29.7 \\
\hline
\end{array}$$
(a) Find the largest time interval over which the percentage growth in the US price per watt of a solar panel was a decreasing function of time. Interpret what decreasing means, practically speaking, in this case.
(b) Find the largest time interval over which the actual price per watt of a solar panel was a decreasing function of time. Interpret what decreasing means, practically speaking, in this case.

Kian M.

Numerade Educator

Problem 65

Table 1.4 shows the average annual sea level, $S$, in meters, in Aberdeen, Scotland, ${ }^{15}$ as a function of time, $t$, measured in years before 2008 .
$$\begin{array}{c|c|c|c|c|c|c}
\hline t & 0 & 25 & 50 & 75 & 100 & 125 \\
\hline S & 7.094 & 7.019 & 6.992 & 6.965 & 6.938 & 6.957 \\
\hline
\end{array}$$
(a) What was the average sea level in Aberdeen in $2008 ?$
(b) In what year was the average sea level 7.019 meters? 6.957 meters?
(c) Table 1.5 gives the average sea level, $S,$ in $A b-$ erdeen as a function of the year, $x$. Complete the missing values.
$$\begin{array}{c|c|c|c|c|c|c}
\hline x & 1883 & ? & 1933 & 1958 & 1983 & 2008 \\
\hline S & ? & 6.938 & ? & 6.992 & ? & ? \\
\hline
\end{array}$$

Alex R.

Numerade Educator

Problem 66

The table gives the required standard weight, $w,$ in kilograms, of American soldiers, aged between 21 and 27 , for height, $h,$ in centimeters. $^{16}$
(a) How do you know that the data in this table could represent a linear function?
(b) Find weight, $w,$ as a linear function of height, $h$ What is the slope of the line? What are the units for the slope?
(c) Find height, $h,$ as a linear function of weight, $w$ What is the slope of the line? What are the units for the slope?
$$\begin{array}{l|c|c|c|c|c|c|c}
\hline h(\mathrm{cm}) & 172 & 176 & 180 & 184 & 188 & 192 & 196 \\
\hline w(\mathrm{kg}) & 79.7 & 82.4 & 85.1 & 87.8 & 90.5 & 93.2 & 95.9 \\
\hline
\end{array}$$

Kian M.

Numerade Educator

Problem 67

A company rents cars at $\$ 40$ a day and 15 cents a mile. Its competitor's cars are $\$ 50$ a day and 10 cents a mile.
(a) For each company, give a formula for the cost of renting a car for a day as a function of the distance traveled.
(b) On the same axes, graph both functions.
(c) How should you decide which company is cheaper?

Natalie A.

Numerade Educator

Problem 68

A $\$ 25,000$ vehicle depreciates $\$ 2000$ a year as it ages. Repair costs are $\$ 1500$ per year.
(a) Write formulas for each of the two linear functions at time $t,$ value, $V(t),$ and repair costs to date, $C(t)$ Graph them.
(b) One strategy is to replace a vehicle when the total cost of repairs is equal to the current value. Find this time.
(c) Another strategy is to replace the vehicle when the value of the vehicle is some percent of the original value. Find the time when the value is $6 \% .$

Alex R.

Numerade Educator

Problem 69

A bakery owner knows that customers buy a total of $q$ cakes when the price, $p,$ is no more than $p=d(q)=$ $20-q / 20$ dollars. She is willing to make and supply as many as $q$ cakes at a price of $p=s(q)=11+q / 40$ dollars each. (The graphs of the functions $d(q)$ and $s(q)$ are called a demand curve and a supply curve, respectively.) The graphs of $d(q)$ and $s(q)$ are in Figure 1.22
(a) Why, in terms of the context, is the slope of $d(q)$ negative and the slope of $s(q)$ positive?
(b) Is each of the ordered pairs $(q, p)$ a solution to the inequality $p \leq 20-q / 20 ?$ Interpret your answers in terms of the context.
\[
(60,18)
\]
(c) Graph in the $q p$ -plane the solution set of the system of inequalities $p \leq 20-q / 20, p \geq 11+q / 40$ What does this solution set represent in terms of the context?
(d) What is the rightmost point of the solution set you graphed in part (c)? Interpret your answer in terms of the context.

Alex R.

Numerade Educator

Problem 70

(a) Consider the functions graphed in Figure $1.23(\mathrm{a})$ Find the coordinates of $C$.
(b) Consider the functions in Figure $1.23(b) .$ Find the coordinates of $C$ in terms of $b$.

Alex R.

Numerade Educator

Problem 71

When Galileo was formulating the laws of motion, he considered the motion of a body starting from rest and falling under gravity. He originally thought that the velocity of such a falling body was proportional to the distance it had fallen. What do the experimental data in Table 1.6 tell you about Galileo's hypothesis? What alternative hypothesis is suggested by the two sets of data in Table 1.6 and Table $1.7 ?$
Table 1.6
$$\begin{array}{c|c|c|c|c|c}
\hline \text { Distance (ft) } & 0 & 1 & 2 & 3 & 4 \\
\hline \text { Velocity (ft/sec) } & 0 & 8 & 11.3 & 13.9 & 16 \\
\hline
\end{array}$$
Table 1.7
$$\begin{array}{c|c|c|c|c|c}
\hline \text { Time (sec) } & 0 & 1 & 2 & 3 & 4 \\
\hline \text { Velocity (ft/sec) } & 0 & 32 & 64 & 96 & 128 \\
\hline
\end{array}$$

Carson M.

Numerade Educator

Problem 72

Explain what is wrong with the statement.
For constants $m$ and $b$, the slope of the linear function $y=b+m x$ is $m=\Delta x / \Delta y$

Gregory H.

Numerade Educator

Problem 73

Explain what is wrong with the statement.
The lines $x=3$ and $y=3$ are both linear functions of $x$

Gregory H.

Numerade Educator

Problem 74

Explain what is wrong with the statement.
The line $y-3=0$ has slope 1 in the $x y$ -plane.

Gregory H.

Numerade Educator

Problem 75

Explain what is wrong with the statement.
Values of $y$ on the graph of $y=0.5 x-3$ increase more slowly than values of $y$ on the graph of $y=0.5-3 x$

Gregory H.

Numerade Educator

Problem 76

Explain what is wrong with the statement.
The equation $y=2 x+1$ indicates that $y$ is directly proportional to $x$ with a constant of proportionality 2 .

Alex R.

Numerade Educator

Problem 77

Give an example of:
A linear function with a positive slope and a negative $x$ -intercept.

Gregory H.

Numerade Educator

Problem 78

Give an example of:
A formula representing the statement " $q$ is inversely proportional to the cube root of $p$ and has a positive constant of proportionality."

Gregory H.

Numerade Educator

Problem 79

True or false? Give an explanation for your answer.
For any two points in the plane, there is a linear function whose graph passes through them.

Gregory H.

Numerade Educator

Problem 80

True or false? Give an explanation for your answer.
If $y=f(x)$ is a linear function, then increasing $x$ by 1 unit changes the corresponding $y$ by $m$ units, where $m$ is the slope.

Gregory H.

Numerade Educator

Problem 81

True or false? Give an explanation for your answer.
The linear functions $y=-x+1$ and $x=-y+1$ have the same graph.

Gregory H.

Numerade Educator

Problem 82

True or false? Give an explanation for your answer.
The linear functions $y=2-2 x$ and $x=2-2 y$ have the same graph.

Gregory H.

Numerade Educator

Problem 83

True or false? Give an explanation for your answer.
If $y$ is a linear function of $x,$ then the ratio $y / x$ is constant for all points on the graph at which $x \neq 0$.

Gregory H.

Numerade Educator

Problem 84

True or false? Give an explanation for your answer.
If $y=f(x)$ is a linear function, then increasing $x$ by 2 units adds $m+2$ units to the corresponding $y$, where $m$ is the slope.

Gregory H.

Numerade Educator

Problem 85

Which of the following functions has its domain identical with its range?
(a) $f(x)=x^{2}$
(b) $g(x)=\sqrt{x}$
(c) $\quad h(x)=x^{3}$
(d) $\quad i(x)=|x|$

Gregory H.

Numerade Educator