1.1E: Describing Relationships Between Two Quantities With Functions (Exercises)
Section 1.1 Exercises
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Select all of the following tables which represent \(y\) as a function of \(x\).
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Select all of the following tables which represent \(y\) as a function of \(x\).
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Select all of the following tables which represent \(y\) as a function of \(x\).
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Select all of the following tables which represent \(y\) as a function of \(x\).
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Select all of the following graphs which represent \(y\) as a function of \(x\).
a b c
d e f -
Select all of the following graphs which represent \(y\) as a function of \(x\).
a b c
d e f - Is the sales tax on an item in California a function of the price of the item?
- Is the daily high temperature of Clovis, California a function of the date?
- Is the date a function of the daily high temperature of Clovis, California?
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Determine if the following equations represent a function
- \(5x-2y=10\)
- \(3x^2+2y^2=1\)
- \(3x-\sqrt{y}=10\)
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Determine if the following equations represent a function
- \(3x+5y=10\)
- \(y^2=x\)
- \(\dfrac{3x}{y-2}=3\)
- Given the graph of the function \(f(x)\) below (a) Evaluate \(f(1)\) (b) Evaluate \(f(3)\) (c) Find all values of x where \(f(x)=3\)
- Given the graph of the function \(g(x)\) below (a) Evaluate \(g(1)\) (b) Evaluate \(g(3)\) (c) Find all values of x where \(g(x)=3\)
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Given the function \(g(x)\) graphed here,
a. Evaluate \(g(2)\)
b. Solve \(g(x) = 2\)
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Given the function \(f(x)\) graphed here,
a. Evaluate \(f(4)\)
b. Solve \(f(x) = 4\)
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Based on the table below,
a. Evaluate \(f(3)\) b. Solve \(f(x) = 1\)\(x\) 0 1 2 3 4 5 6 7 8 9 \(f(x)\) 74 28 1 53 56 3 36 45 14 47 -
Based on the table below,
a. Evaluate \(f(8)\) b. Solve \(f(x) = 1\)\(x\) 0 1 2 3 4 5 6 7 8 9 \(f(x)\) 62 8 7 38 86 73 70 39 75 34 -
Let \(f(x) = 4 - 2x\) and \(g(x) = 8x^2 - 7x + 3\).
- Evaluate \(f(2)\)
- Evaluate \(g(2)\)
- Evaluate \(f(-1)\)
- Evaluate \(g(-1)\)
- Find all values of \(x\) where \(f(x)=0\)
- Evaluate \(f(5)-f(3)\)
- Evaluate \(\dfrac{g(2)-g(1)}{2-1}\)
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Let \(h(x) = 8 - 3x\) and \(k(x) = 6x^2 - 7x + 4\).
- Evaluate \(h(3)\)
- Evaluate \(k(3)\)
- Evaluate \(h(-2)\)
- Evaluate \(k(-2)\)
- Find all values of \(x\) where \(h(x)=0\)
- Evaluate \(h(4)+h(1)\)
- Evaluate \(\dfrac{k(4)-k(1)}{4-1}\)
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Let \(f(x) = -x^3 + 2x\) and \(g(x) = 3 + \sqrt{x + 3}\)
- Evaluate \(f(4)\)
- Evaluate \(g(4)\)
- Evaluate \(f(-4)\)
- Evaluate \(g(-4)\)
- Find all values of \(x\) where \(g(x)=0\)
- Evaluate \(f(6)-f(1)\)
- Evaluate \(\dfrac{g(6)-g(1)}{6-1}\)
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Let \(m(x) = 5x^4 + x^2\) and \(n(x) = 4 - \sqrt[3]{x - 2}\)
- Evaluate \(m(3)\)
- Evaluate \(n(3)\)
- Evaluate \(m(-2)\)
- Evaluate \(n(2)\)
- Find all values of \(x\) where \(n(x)=0\)
- Evaluate \(n(10)+n(1)\)
- Evaluate \(\dfrac{m(4)-m(1)}{4-1}\)
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Let \(f(x) = \dfrac{x - 3}{x + 1}\)
- Evaluate \(f(4)\)
- Evaluate \(f(0)\)
- Evaluate \(f(-1)\)
- Find all values of \(x\) where \(f(x)=0\)
- Evaluate \(f(4)+f(2)\)
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Let \(f(x) = 3x+2\). Evaluate and simplify each of the following
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\(f(a)\)
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\(f(a+h)\)
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\(\dfrac{f(x)-f(a)}{x-a}\)
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\(\dfrac{f(a+h)-f(a)}{h}\)
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Let \(g(x) = x^2+7\). Evaluate and simplify each of the following
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\(g(a)\)
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\(g(a+h)\)
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\(\dfrac{g(x)-g(a)}{x-a}\)
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\(\dfrac{g(a+h)-g(a)}{h}\)
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Let \(f(x) = x^2+5x\). Evaluate and simplify each of the following
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Evaluate \(f(a+h)\)
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Evaluate \(\dfrac{f(a+h)-f(a)}{h}\)
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Let \(f(x) = 3x^2-4x+2\).
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Evaluate and simplify \(\dfrac{f(a+h)-f(a)}{h}\)
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- Let \(f(x) = \dfrac{1}{x}\). Evaluate and simplify \(\dfrac{f(x)-f(a)}{x-a}\).
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Let \(f(t)\) be the number of ducks in a lake
t
years after 1990. Explain the meaning of each statement:
a. \(f\left(5\right)=30\) b. \(f\left(10\right)=40\) -
Let \(h(t)\) be the height above ground, in feet, of a rocket
t
seconds after launching. Explain the meaning of each statement:
a. \(h(1)=200\) b. \(h(2)=350\) -
The amount of garbage,
G
, produced by a city with population
p
is given by \(G = f(p)\).
G
is measured in tons per week, and \(p\) is measured in thousands of people. a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\).
b. Explain the meaning of the statement \(f(5)=2\). -
The number of cubic yards of dirt,
D
, needed to cover a garden with area
a
square feet is given by \(D=g(a)\). a. A garden with area 5000 ft \({}^{2}\) requires 50 cubic yards of dirt. Express this information in terms of the function \(g\).
b. Explain the meaning of the statement \(g(100)=1\). -
Dave leaves his office in Padelford Hall on his way to teach in Gould Hall. Below are several different scenarios. In each case, sketch a plausible (reasonable) graph of the function \(s = d(t)\) which keeps track of Dave’s distance
s
from Padelford Hall at time \(t\). Take distance units to be “feet” and time units to be “minutes.” Assume Dave’s path to Gould Hall is long a straight line which is 2400 feet long. [UW]
a. Dave leaves Padelford Hall and walks at a constant spend until he reaches Gould Hall 10 minutes later.
b. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for 1 minute. He then continues on to Gould Hall at the same constant speed he had when he originally left Padelford Hall.
c. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for 1 minute to figure out where he is. Dave then continues on to Gould Hall at twice the constant speed he had when he originally left Padelford Hall.
d. Dave leaves Padelford Hall and walks at a constant speed. It takes him 6 minutes to reach the half-way point. Then he gets confused and stops for 1 minute to figure out where he is. Dave is totally lost, so he simply heads back to his office, walking the same constant speed he had when he originally left Padelford Hall.
e. Dave leaves Padelford heading for Gould Hall at the same instant Angela leaves Gould Hall heading for Padelford Hall. Both walk at a constant speed, but Angela walks twice as fast as Dave. Indicate a plot of “distance from Padelford” vs. “time” for the both Angela and Dave.
f. Suppose you want to sketch the graph of a new function \(s = g(t)\) that keeps track of Dave’s distance s from Gould Hall at time t. How would your graphs change in (a) - (e)? -
Match each story about a bike ride with the appropriate graph, where \(d\) represents the distance from home \(t\) hours since the start of the ride.
- The rider starts from home and rides 5 miles per hour away from home
- The rider starts 5 miles from home and rides 5 miles per hour away from home.
- The rider starts 5 miles from home and rides 10 miles per hour away from home.
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The rider starts 10 miles from home and rides 5 miles per hour towards their home.
- III. V.
- VI.
- Answer
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1. Tables (a) and (b) represent \(y\) as a function of \(x\) because for every value of \(x\) there is only one value for \(y\). Table (c) is not a function because for the input \(x = 10\), there are two different outputs for \(y\).
3. Tables (a) (b) and (d) represent \(y\) as a function of \(x\) because for every value of \(x\) there is only one value for \(y\). Table (c) is not a function because for the input \(x=3\), there are two different outputs for \(y\). Even though the input \(x = 3\) is listed twice in table (a), it is associated with only one output \(y =1\).
5. Graphs (a) (b) (d) and (e) represent \(y\) as a function of \(x\) because for every value of \(x\) there is only one value for \(y\). Graphs (c) and (f) are not functions because they contain points that have more than one output for a given input, or values for \(x\) that have 2 or more values for \(y\).
7. Yes, the sales tax on an item in California a function of the price of the item. For each price, there is exactly one tax charged since sales tax is typically a percentage of the price.
9. No, the date a function of the daily high temperature of Clovis, California. Two different days may have the same daily high temperature.
11. (a) Yes, the equation does represent a function since the equation can be written as a single formula for \(y\) where \(y=\dfrac{-3}{5}x+2\).
(b) No, the equation doesn't represent a function since the equation can't be written as a single formula for \(y\) where \(y=\pm \sqrt{x}\).
(c) (a) Yes, the equation does represent a function since the equation can be written as a single formula for \(y\) where \(y=\dfrac{3}{2}x+2\).
13. (a) \(g(1)=3\) (b) \(g(3)=-1\) (c) \(x=1\) and \(x=1\)
15. (a) \(f(4) = 1\) (b) \(x = -2\)
17. (a) \(f(8) = 75\) (b) There is no solution to \(f(x) = 1\)
19. (a) \(h(3)-1\) (b) \(k(3)=37\) (c) \(h(-2)= 14\) (d) \(k(-2)=42\) (e) \(x=\dfrac{-8}{3}\) (f) \(h(4)+h(1)=1\) (g) \(\dfrac{k(4)-k(1)}{4-1}=23\)
21. (a) \(m(3)=414\) (b) \(n(3)=3\) (c) \(m(-2)=84\) (d) \(n(2)=4\) (e) \(x=66\) (f) \(n(10)+n(1)=7\) (g) \(\dfrac{m(4)-m(1)}{4-1}=430\)
23.(a) \(f(a)=3a+2\) (b) \(f(a+h)=3a+3h+2\) (c) \(\dfrac{f(x)-f(a)}{x-a}= 3\) (d) \(\dfrac{f(a+h)-f(a)}{h} =3 \)
25. (a) \(f(a+h)=a^2+2ah+h^2+5a+5h\) (b) \(\dfrac{g(a+h)-g(a)}{h} = 2a+h+5\)
27. \(\dfrac{f(x)-f(a)}{x-a}=\dfrac{-1}{ax}\)
29. (a) After 1 second, the height of the rocket is 200 ft. (b) After 2 seconds, the height of the rocket is 350 ft.
31. (a) \(g(5000)=50\) (b) A garden with area 100 ft \({}^{2}\) requires 1 cubic yards of dirt.
33. (a) Graph II (b) Graph V (c) Graph I (d) Graph III