MAT170 - Exam 1 Review - Fall 2022 Name: _________________________________
1. Given h( x) = x3 − x + 10 and g ( x) = 2x + 2, find h( g ( x)) and h( g (1)) .
2. A linear relationship exists between the quantities whose values are represented by x and y.
(a) What is the value of y when x is 4? x y
-1 5
(b) Write an equation that expresses Δy in terms of Δx 2 14
4
(c) Write an equation that expresses y in terms of x. 8 32
(d) Is y proportional to x in this relationship ?
(e) Is Δy proportional to Δx in this relationship ?
3. Paulo is running along the beach at a constant rate of 3 ft/sec.
(a) How many feet does Paulo travel in 11.8 sec?
(b) How many seconds (rounded to the nearest hundredth) will it take for Paulo to travel 132 feet?
(c) Suppose Paulo started running when he was 20 feet from the boardwalk, and he ran in a straight
line away from the boardwalk and towards the snack bar. Write a formula that determines Paulo’s
distance d from the boardwalk (in feet), given the amount of time t (in seconds) since Paulo
started running.
(d) If the distance from the boardwalk to the snack bar is 1000 feet, write a formula that determines
Paulo’s distance s from the snack bar (in feet), given the amount of time t (in seconds) since
Paulo started running from the boardwalk.
(e) Since Paulo starts running when he is 20 feet from the boardwalk, and he runs at a constant speed
of 3 ft/sec, is it true that the total distance d that Paulo is from the boardwalk after t seconds of
running will be 3 times the number of seconds that he has been running? Why or why not?
,MAT170 - Exam 1 Review - Fall 2022 Name: _________________________________
4. On a sunny winter day in Illinois, a spherical snowball has a radius of 10 cm and as it melts, the radius
4
decreases at a rate of 0.4 cm/minute. Recall that the volume of a sphere is determined by 𝑉 = 𝜋𝑟 3
3
(a) What is the volume of the snowball 5.3 minutes after it begins melting?
(b) Write a formula that expresses the radius of the snowball (in cm) in terms of the number of
minutes that it has been melting.
(c) Define a function g that expresses the volume of the snowball (in cm3) as a function of the
amount of time (in minutes) since the snowball began melting.
5. The height of a boomerang thrown in the air can be modeled by the function:
h = f (t) = -.63t 2 + 3.9t + 5
where h is the height of the boomerang (in feet) above the ground t seconds after being thrown.
(a) What is the average speed of the boomerang as it travels during the time interval from 2 seconds
to 3 seconds after it was thrown?
(b) Write a statement that describes the meaning of average speed in this context.
6. Suppose the function g represents the cost of tuition at ASU t years since 2000, and that tuition has
been rising each year since the year 2000.
(a) What does the expression 3 g(6) represent ?
(b) What does the expression g −1 (15000) represent ?
Use the table below to answer question 7 :
x -2 -1 0 1 2
f(x) 2 0 1 -2 -1
g(x) 0 2 -1 -2 1
7. Determine (
a. g f ( 2 ) ) b. g(f(-2)) c. f
−1
(g ( 0) ) (
d. f −1 g −1 ( 0) )
, MAT170 - Exam 1 Review - Fall 2022 Name: _________________________________
8. Let d be the distance in feet of a race car from the starting line, and let t be the number of seconds
since the race began. The distance d is a function of the time t and is represented by the function f,
defined by : d = f(t) = 3.2t2 + 3t.
(a) What is the meaning of f(7) – f(3)?
(b) Compute the race car’s average speed over the interval t = 3 to t = 7 sec.
(c) Write a statement that best describes the meaning of average speed in this context?
9. What is the domain of the following functions. (Write your answers in interval notation.)
x-3 5
(a) n(x) = (b) 𝑔(𝑡) = 𝑡 2 −9 (c) ℎ(𝑎) = √2𝑎 + 7
x -10
10. A bucket is filled with water up to the 7 gallon mark. The bucket springs a leak and water begins draining at a
constant rate of 3/8 gallon per minute.
(a) Write a function that determines the number of gallons of water n in the bucket in terms of the
number of minutes t the water has been draining.
(b) Is n proportional to t in this relationship ?
(c) Is Δn proportional to Δt in this relationship ?
(d) If d represents the number of gallons of water that have drained from the tank t minutes
after the water began draining, is d proportional to t in this scenario ?
11. Lisa and Sarah decided to meet at a park bench near both of their homes. Lisa lives 2390 feet due west of the
park bench and Sarah lives 1420 feet due east of the park bench. Sarah left her house at 7:00 pm and traveled at
a constant speed of 210 feet per minute towards the bench. Lisa left her house at the same time traveling a
constant speed of 330 feet per minute towards the bench.
(a) Determine the function f (t) that expresses the number of feet, d, between Lisa and Sarah in terms of
the number of minutes, t, since they started walking toward one another.
(b) Is d proportional to t in this situation ?
(c) Determine the function f −1 (d) that expresses the number of minutes, t, since Lisa and Sarah started
walking, in terms of the number of feet d between them.
1. Given h( x) = x3 − x + 10 and g ( x) = 2x + 2, find h( g ( x)) and h( g (1)) .
2. A linear relationship exists between the quantities whose values are represented by x and y.
(a) What is the value of y when x is 4? x y
-1 5
(b) Write an equation that expresses Δy in terms of Δx 2 14
4
(c) Write an equation that expresses y in terms of x. 8 32
(d) Is y proportional to x in this relationship ?
(e) Is Δy proportional to Δx in this relationship ?
3. Paulo is running along the beach at a constant rate of 3 ft/sec.
(a) How many feet does Paulo travel in 11.8 sec?
(b) How many seconds (rounded to the nearest hundredth) will it take for Paulo to travel 132 feet?
(c) Suppose Paulo started running when he was 20 feet from the boardwalk, and he ran in a straight
line away from the boardwalk and towards the snack bar. Write a formula that determines Paulo’s
distance d from the boardwalk (in feet), given the amount of time t (in seconds) since Paulo
started running.
(d) If the distance from the boardwalk to the snack bar is 1000 feet, write a formula that determines
Paulo’s distance s from the snack bar (in feet), given the amount of time t (in seconds) since
Paulo started running from the boardwalk.
(e) Since Paulo starts running when he is 20 feet from the boardwalk, and he runs at a constant speed
of 3 ft/sec, is it true that the total distance d that Paulo is from the boardwalk after t seconds of
running will be 3 times the number of seconds that he has been running? Why or why not?
,MAT170 - Exam 1 Review - Fall 2022 Name: _________________________________
4. On a sunny winter day in Illinois, a spherical snowball has a radius of 10 cm and as it melts, the radius
4
decreases at a rate of 0.4 cm/minute. Recall that the volume of a sphere is determined by 𝑉 = 𝜋𝑟 3
3
(a) What is the volume of the snowball 5.3 minutes after it begins melting?
(b) Write a formula that expresses the radius of the snowball (in cm) in terms of the number of
minutes that it has been melting.
(c) Define a function g that expresses the volume of the snowball (in cm3) as a function of the
amount of time (in minutes) since the snowball began melting.
5. The height of a boomerang thrown in the air can be modeled by the function:
h = f (t) = -.63t 2 + 3.9t + 5
where h is the height of the boomerang (in feet) above the ground t seconds after being thrown.
(a) What is the average speed of the boomerang as it travels during the time interval from 2 seconds
to 3 seconds after it was thrown?
(b) Write a statement that describes the meaning of average speed in this context.
6. Suppose the function g represents the cost of tuition at ASU t years since 2000, and that tuition has
been rising each year since the year 2000.
(a) What does the expression 3 g(6) represent ?
(b) What does the expression g −1 (15000) represent ?
Use the table below to answer question 7 :
x -2 -1 0 1 2
f(x) 2 0 1 -2 -1
g(x) 0 2 -1 -2 1
7. Determine (
a. g f ( 2 ) ) b. g(f(-2)) c. f
−1
(g ( 0) ) (
d. f −1 g −1 ( 0) )
, MAT170 - Exam 1 Review - Fall 2022 Name: _________________________________
8. Let d be the distance in feet of a race car from the starting line, and let t be the number of seconds
since the race began. The distance d is a function of the time t and is represented by the function f,
defined by : d = f(t) = 3.2t2 + 3t.
(a) What is the meaning of f(7) – f(3)?
(b) Compute the race car’s average speed over the interval t = 3 to t = 7 sec.
(c) Write a statement that best describes the meaning of average speed in this context?
9. What is the domain of the following functions. (Write your answers in interval notation.)
x-3 5
(a) n(x) = (b) 𝑔(𝑡) = 𝑡 2 −9 (c) ℎ(𝑎) = √2𝑎 + 7
x -10
10. A bucket is filled with water up to the 7 gallon mark. The bucket springs a leak and water begins draining at a
constant rate of 3/8 gallon per minute.
(a) Write a function that determines the number of gallons of water n in the bucket in terms of the
number of minutes t the water has been draining.
(b) Is n proportional to t in this relationship ?
(c) Is Δn proportional to Δt in this relationship ?
(d) If d represents the number of gallons of water that have drained from the tank t minutes
after the water began draining, is d proportional to t in this scenario ?
11. Lisa and Sarah decided to meet at a park bench near both of their homes. Lisa lives 2390 feet due west of the
park bench and Sarah lives 1420 feet due east of the park bench. Sarah left her house at 7:00 pm and traveled at
a constant speed of 210 feet per minute towards the bench. Lisa left her house at the same time traveling a
constant speed of 330 feet per minute towards the bench.
(a) Determine the function f (t) that expresses the number of feet, d, between Lisa and Sarah in terms of
the number of minutes, t, since they started walking toward one another.
(b) Is d proportional to t in this situation ?
(c) Determine the function f −1 (d) that expresses the number of minutes, t, since Lisa and Sarah started
walking, in terms of the number of feet d between them.
