Stewart Precalc 6e Section 2.6: Combining Functions

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Stewart Precalc 6e Section 2.6: Combining Functions 0/70
1 Concepts - Combining functions from graph · Level 1
From the graphs of \(f\) and \(g\) in the figure, find (a) \((f+g)(2)\), (b) \((f-g)(2)\), (c) \((f g)(2)\), (d) \(\left(\dfrac{f}{g}\right)(2)\).
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2 Concepts - Composition definition · Level 1
By definition, \((f \circ g)(x) = \) ____. So if \(g(2) = 5\) and \(f(5) = 12\), then \((f \circ g)(2) = \) ____.
3 Concepts - Composition rules · Level 1
If the rule of the function \(f\) is 'add one' and the rule of the function \(g\) is 'multiply by 2,' then state the rule of \(f \circ g\) and the rule of \(g \circ f\).
4 Concepts - Composition algebra · Level 1
Express the functions in Exercise 3 algebraically: \(f(x) = \) ____, \(g(x) = \) ____, \((f \circ g)(x) = \) ____, \((g \circ f)(x) = \) ____.
5 Sum, difference, product, quotient · Level 2
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = x - 3\), \(g(x) = x^2\).
6 Sum, difference, product, quotient · Level 2
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = x^2 + 2x\), \(g(x) = 3x^2 - 1\).
7 Sum, difference, product, quotient · Level 2
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = \sqrt{4 - x^2}\), \(g(x) = \sqrt{1 + x}\).
8 Sum, difference, product, quotient · Level 2
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = \sqrt{9 - x^2}\), \(g(x) = \sqrt{x^2 - 4}\).
9 Sum, difference, product, quotient · Level 2
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = \dfrac{2}{x}\), \(g(x) = 4/(x + 4)\).
10 Sum, difference, product, quotient · Level 2
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = 2/(x + 1)\), \(g(x) = x/(x + 1)\).
11 Domain · Level 2
Find the domain of the function \(f(x) = \sqrt{x} + \sqrt{1 - x}\).
12 Domain · Level 2
Find the domain of the function \(g(x) = \sqrt{x + 1} - \dfrac{1}{x}\).
13 Domain · Level 2
Find the domain of the function \(h(x) = (x - 3)^{-\dfrac{1}{4}}\).
14 Domain · Level 2
Find the domain of the function \(k(x) = \dfrac{\sqrt{x + 3}}{x - 1}\).
15 Graphical addition · Level 2
Use graphical addition to sketch the graph of \(f + g\) from the figure.
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16 Graphical addition · Level 2
Use graphical addition to sketch the graph of \(f + g\) from the figure.
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17 Graph addition with formula · Level 2
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen to illustrate graphical addition for \(f(x) = \sqrt{1 + x}\), \(g(x) = \sqrt{1 - x}\).
18 Graph addition with formula · Level 2
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen for \(f(x) = x^2\), \(g(x) = \sqrt{x}\).
19 Graph addition with formula · Level 2
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen for \(f(x) = x^2\), \(g(x) = \left(\dfrac{1}{3}\right) x^3\).
20 Graph addition with formula · Level 2
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen for \(f(x) = \sqrt[4]{1 - x}\), \(g(x) = \sqrt{1 - x^2/9}\).
21 Composition evaluation · Level 2
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \(f(g(0))\), (b) \(g(f(0))\).
22 Composition evaluation · Level 2
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \(f(f(4))\), (b) \(g(g(3))\).
23 Composition evaluation · Level 2
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \((f \circ g)(-2)\), (b) \((g \circ f)(-2)\).
24 Composition evaluation · Level 2
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \((f \circ f)(-1)\), (b) \((g \circ g)(2)\).
25 Composition formula · Level 3
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to find (a) \((f \circ g)(x)\), (b) \((g \circ f)(x)\).
26 Composition formula · Level 3
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to find (a) \((f \circ f)(x)\), (b) \((g \circ g)(x)\).
27 Composition from graphs · Level 2
Use the given graphs of \(f\) and \(g\) to evaluate \(f(g(2))\).
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28 Composition from graphs · Level 2
Use the given graphs of \(f\) and \(g\) to evaluate \(g(f(0))\).
29 Composition from graphs · Level 2
Use the given graphs of \(f\) and \(g\) to evaluate \((g \circ f)(4)\).
30 Composition from graphs · Level 2
Use the given graphs of \(f\) and \(g\) to evaluate \((f \circ g)(0)\).
31 Composition from graphs · Level 2
Use the given graphs of \(f\) and \(g\) to evaluate \((g \circ g)(-2)\).
32 Composition from graphs · Level 2
Use the given graphs of \(f\) and \(g\) to evaluate \((f \circ f)(4)\).
33 Find compositions · Level 2
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = 2x + 3\), \(g(x) = 4x - 1\).
34 Find compositions · Level 2
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = 6x - 5\), \(g(x) = \dfrac{x}{2}\).
35 Find compositions · Level 2
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x^2\), \(g(x) = x + 1\).
36 Find compositions · Level 2
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x^3 + 2\), \(g(x) = \sqrt[3]{x}\).
37 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = \dfrac{1}{x}\), \(g(x) = 2x + 4\).
38 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x^2\), \(g(x) = \sqrt{x - 3}\).
39 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = |x|\), \(g(x) = 2x + 3\).
40 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x - 4\), \(g(x) = |x + 4|\).
41 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x/(x + 1)\), \(g(x) = 2x - 1\).
42 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = \dfrac{1}{\sqrt{x}}\), \(g(x) = x^2 - 4x\).
43 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x/(x + 1)\), \(g(x) = \dfrac{1}{x}\).
44 Find compositions · Level 3
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = \dfrac{2}{x}\), \(g(x) = x/(x + 2)\).
45 Triple composition · Level 3
Find \(f \circ g \circ h\) for \(f(x) = x - 1\), \(g(x) = \sqrt{x}\), \(h(x) = x - 1\).
46 Triple composition · Level 3
Find \(f \circ g \circ h\) for \(f(x) = \dfrac{1}{x}\), \(g(x) = x^3\), \(h(x) = x^2 + 2\).
47 Triple composition · Level 3
Find \(f \circ g \circ h\) for \(f(x) = x^4 + 1\), \(g(x) = x - 5\), \(h(x) = \sqrt{x}\).
48 Triple composition · Level 3
Find \(f \circ g \circ h\) for \(f(x) = \sqrt{x}\), \(g(x) = x/(x - 1)\), \(h(x) = \sqrt[3]{x}\).
49 Express as composition · Level 2
Express the function \(F(x) = (x - 9)^5\) in the form \(f \circ g\).
50 Express as composition · Level 2
Express the function \(F(x) = \sqrt{x} + 1\) in the form \(f \circ g\).
51 Express as composition · Level 2
Express the function \(G(x) = x^2/(x^2 + 4)\) in the form \(f \circ g\).
52 Express as composition · Level 2
Express the function \(G(x) = 1/(x + 3)\) in the form \(f \circ g\).
53 Express as composition · Level 2
Express the function \(H(x) = |1 - x^3|\) in the form \(f \circ g\).
54 Express as composition · Level 2
Express the function \(H(x) = \sqrt{1 + \sqrt{x}}\) in the form \(f \circ g\).
55 Express as triple composition · Level 3
Express the function \(F(x) = 1/(x^2 + 1)\) in the form \(f \circ g \circ h\).
56 Express as triple composition · Level 3
Express the function \(F(x) = \sqrt[3]{\sqrt{x} - 1}\) in the form \(f \circ g \circ h\).
57 Express as triple composition · Level 3
Express the function \(G(x) = (4 + \sqrt[3]{x})^9\) in the form \(f \circ g \circ h\).
58 Express as triple composition · Level 3
Express the function \(G(x) = 2/(3 + \sqrt{x})^2\) in the form \(f \circ g \circ h\).
59 Applications - Revenue · Level 3
A print shop makes bumper stickers for election campaigns. If \(x\) stickers are ordered (where \(x < 10000\)), then the price per bumper sticker is \(0.15 - 0.000002 x\) dollars, and the total cost of producing the order is \(0.095 x - 0.0000005 x^2\) dollars. Use the fact that revenue \(=\) price per item \(\times\) number of items sold to express \(R(x)\), the revenue from an order of \(x\) stickers, as a product of two functions of \(x\).
60 Applications - Profit · Level 3
Use the fact that profit \(=\) revenue \(-\) cost to express \(P(x)\), the profit on an order of \(x\) stickers, as a difference of two functions of \(x\). (See Exercise 59.)
61 Applications - Area of a Ripple · Level 3
A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of \(60\) cm/s. (a) Find a function \(g\) that models the radius as a function of time. (b) Find a function \(f\) that models the area of the circle as a function of the radius. (c) Find \(f \circ g\). What does this function represent?
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62 Applications - Inflating a Balloon · Level 3
A spherical balloon is being inflated. The radius of the balloon is increasing at the rate of \(1\) cm/s. (a) Find a function \(f\) that models the radius as a function of time. (b) Find a function \(g\) that models the volume as a function of the radius. (c) Find \(g \circ f\). What does this function represent?
63 Applications - Area of a Balloon · Level 3
A spherical weather balloon is being inflated. The radius of the balloon is increasing at the rate of \(2\) cm/s. Express the surface area of the balloon as a function of time \(t\) (in seconds).
64 Applications - Multiple Discounts · Level 3
You have a \(\$50\) coupon from the manufacturer good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a \(20%\) discount on all cell phones. Let \(x\) represent the regular price of the cell phone. (a) Suppose only the \(20%\) discount applies. Find a function \(f\) that models the purchase price. (b) Suppose only the \(\$50\) coupon applies. Find a function \(g\) that models the purchase price. (c) Find both \((f \circ g)(x)\) and \((g \circ f)(x)\). Which composition gives the lower price?
65 Applications - Multiple Discounts · Level 3
An appliance dealer advertises a \(10%\) discount on all his washing machines. In addition, the manufacturer offers a \(\$100\) rebate on the purchase of a washing machine. Let \(x\) represent the sticker price. (a) Find \(f\) for the \(10%\) discount only. (b) Find \(g\) for the \(\$100\) rebate only. (c) Find \(f \circ g\) and \(g \circ f\). Which is the better deal?
66 Applications - Airplane Trajectory · Level 3
An airplane is flying at a speed of \(350\) mi/h at an altitude of one mile. The plane passes directly above a radar station at time \(t = 0\). (a) Express the distance \(s\) (in miles) between the plane and the radar station as a function of the horizontal distance \(d\). (b) Express \(d\) as a function of time \(t\) (in hours). (c) Use composition to express \(s\) as a function of \(t\).
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67 Discovery - Compound Interest · Level 4
A savings account earns \(5%\) interest compounded annually. If you invest \(x\) dollars, then \(A(x) = x + 0.05 x = 1.05 x\) is the amount after one year. Find \(A \circ A\), \(A \circ A \circ A\), and \(A \circ A \circ A \circ A\). What do these compositions represent? Find a formula for the composition of \(n\) copies of \(A\).
68 Discovery - Composing Linear Functions · Level 4
The graphs of \(f(x) = m_1 x + b_1\) and \(g(x) = m_2 x + b_2\) are lines with slopes \(m_1\) and \(m_2\), respectively. Is the graph of \(f \circ g\) a line? If so, what is its slope?
69 Discovery - Solving for Unknown Function · Level 4
Suppose \(g(x) = 2x + 1\) and \(h(x) = 4x^2 + 4x + 7\). Find a function \(f\) such that \(f \circ g = h\). Now suppose \(f(x) = 3x + 5\) and \(h(x) = 3x^2 + 3x + 2\). Find a function \(g\) such that \(f \circ g = h\).
70 Discovery - Odd and Even Compositions · Level 4
Suppose \(h = f \circ g\). (a) If \(g\) is even, is \(h\) necessarily even? (b) If \(g\) is odd, is \(h\) odd? (c) What if \(g\) is odd and \(f\) is odd? (d) What if \(g\) is odd and \(f\) is even?

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