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Stewart Precalc 6e Section 2.6: Combining Functions
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단원별 정답률
Find compositions
약점
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Sum, difference, product, quotient
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Composition from graphs
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Express as composition
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Domain
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Graph addition with formula
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Composition evaluation
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Triple composition
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Express as triple composition
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Graphical addition
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Composition formula
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Applications - Multiple Discounts
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Concepts - Combining functions from graph
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Concepts - Composition definition
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Concepts - Composition rules
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Concepts - Composition algebra
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Applications - Revenue
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Applications - Profit
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Applications - Area of a Ripple
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Applications - Inflating a Balloon
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Applications - Area of a Balloon
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Applications - Airplane Trajectory
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Discovery - Compound Interest
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Discovery - Composing Linear Functions
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Discovery - Solving for Unknown Function
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Discovery - Odd and Even Compositions
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약점 단원을 집중 연습으로 보강하세요.
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1
Concepts - Combining functions from graph
오답
From the graphs of \(f\) and \(g\) in the figure, find (a) \((f+g)(2)\), (b) \((f-g)(2)\), (c) \((f g)(2)\), (d) \(\left(\dfrac{f}{g}\right)(2)\).
(미작성)
정답
(a) \(f(2) + g(2)\), (b) \(f(2) - g(2)\), (c) \(f(2) \cdot g(2)\), (d) \(f(2)/g(2)\)
위 정답과 비교하여 채점하세요:
해설
Read the values of \(f(2)\) and \(g(2)\) directly from the graphs, then apply the indicated operations: \((f+g)(x) = f(x) + g(x)\), \((f-g)(x) = f(x) - g(x)\), \((f g)(x) = f(x) g(x)\), and \(\left(\dfrac{f}{g}\right)(x) = f(x)/g(x)\).
2
Concepts - Composition definition
오답
By definition, \((f \circ g)(x) = \) ____. So if \(g(2) = 5\) and \(f(5) = 12\), then \((f \circ g)(2) = \) ____.
(미작성)
정답
\((f \circ g)(x) = f(g(x))\); \((f \circ g)(2) = 12\)
위 정답과 비교하여 채점하세요:
해설
By definition \((f \circ g)(x) = f(g(x))\), so \((f \circ g)(2) = f(g(2)) = f(5) = 12\).
3
Concepts - Composition rules
오답
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 \(g \circ f\).
(미작성)
정답
Rule of \(f \circ g\): 'multiply by 2, then add 1.' Rule of \(g \circ f\): 'add 1, then multiply by 2.'
위 정답과 비교하여 채점하세요:
해설
\(f \circ g\) applies \(g\) first (multiply by 2), then applies \(f\) (add 1). \(g \circ f\) applies \(f\) first (add 1), then applies \(g\) (multiply by 2).
4
Concepts - Composition algebra
오답
Express the functions in Exercise 3 algebraically: \(f(x) = \) ____, \(g(x) = \) ____, \((f \circ g)(x) = \) ____, \((g \circ f)(x) = \) ____.
(미작성)
정답
\(f(x) = x + 1\), \(g(x) = 2x\), \((f \circ g)(x) = 2x + 1\), \((g \circ f)(x) = 2x + 2\)
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = f(g(x)) = f(2x) = 2x + 1\). \((g \circ f)(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2\).
5
Sum, difference, product, quotient
오답
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = x - 3\), \(g(x) = x^2\).
(미작성)
정답
\((f+g)(x) = x^2 + x - 3\), domain \(RR\). \((f-g)(x) = -x^2 + x - 3\), domain \(RR\). \((f g)(x) = x^3 - 3x^2\), domain \(RR\). \(\left(\dfrac{f}{g}\right)(x) = (x-3)/x^2\), domain \(x \neq 0\).
위 정답과 비교하여 채점하세요:
해설
Both \(f\) and \(g\) have domain \(RR\). The sum, difference, and product also have domain \(RR\). For the quotient, exclude values where \(g(x) = 0\), i.e., \(x = 0\).
6
Sum, difference, product, quotient
오답
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = x^2 + 2x\), \(g(x) = 3x^2 - 1\).
(미작성)
정답
\((f+g)(x) = 4x^2 + 2x - 1\), \((f-g)(x) = -2x^2 + 2x + 1\), \((f g)(x) = (x^2 + 2x)(3x^2 - 1)\), \(\left(\dfrac{f}{g}\right)(x) = \dfrac{x^2 + 2x}{3x^2 - 1}\). Domain \(RR\) for the first three; \(x \neq \pm \dfrac{\sqrt{3}}{3}\) for the quotient.
위 정답과 비교하여 채점하세요:
해설
Both functions have domain \(RR\). Exclude \(x\) values where \(3x^2 - 1 = 0\), namely \(x = \pm \dfrac{1}{\sqrt{3}} = \pm \dfrac{\sqrt{3}}{3}\), from the quotient's domain.
7
Sum, difference, product, quotient
오답
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = \sqrt{4 - x^2}\), \(g(x) = \sqrt{1 + x}\).
(미작성)
정답
\((f+g)(x) = \sqrt{4 - x^2} + \sqrt{1 + x}\), \((f-g)(x) = \sqrt{4 - x^2} - \sqrt{1 + x}\), \((f g)(x) = \sqrt{(4 - x^2)(1 + x)}\), \(\left(\dfrac{f}{g}\right)(x) = \dfrac{\sqrt{4 - x^2}}{\sqrt{1 + x}}\). Domain \([-1, 2]\) for first three; \((-1, 2]\) for quotient.
위 정답과 비교하여 채점하세요:
해설
Domain of \(f\) is \([-2, 2]\) and of \(g\) is \([-1, \infty)\). The intersection is \([-1, 2]\). For the quotient, exclude \(x = -1\) where \(g(x) = 0\).
8
Sum, difference, product, quotient
오답
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = \sqrt{9 - x^2}\), \(g(x) = \sqrt{x^2 - 4}\).
(미작성)
정답
\((f+g)(x) = \sqrt{9 - x^2} + \sqrt{x^2 - 4}\), similar for \(f-g\), \(f g = \sqrt{(9 - x^2)(x^2 - 4)}\), \(\left(\dfrac{f}{g}\right)(x) = \dfrac{\sqrt{9 - x^2}}{\sqrt{x^2 - 4}}\). Domain \([-3, -2] \cup [2, 3]\); \((-3, -2) \cup (2, 3)\) for quotient.
위 정답과 비교하여 채점하세요:
해설
Domain of \(f\) is \([-3, 3]\) and of \(g\) is \((-\infty, -2] \cup [2, \infty)\). Their intersection is \([-3, -2] \cup [2, 3]\). For the quotient, exclude \(x = \pm 2\) where \(g(x) = 0\).
9
Sum, difference, product, quotient
오답
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = \dfrac{2}{x}\), \(g(x) = 4/(x + 4)\).
(미작성)
정답
\((f+g)(x) = \dfrac{6x + 8}{x(x + 4)}\), \((f-g)(x) = \dfrac{-2x + 8}{x(x + 4)}\), \((f g)(x) = 8/(x(x + 4))\), \(\left(\dfrac{f}{g}\right)(x) = \dfrac{x + 4}{2x}\). Domain: \(x \neq 0, -4\).
위 정답과 비교하여 채점하세요:
해설
Combine over common denominator \(x(x + 4)\). Exclude \(x = 0\) (from \(f\)) and \(x = -4\) (from \(g\)). The quotient simplifies after clearing fractions.
10
Sum, difference, product, quotient
오답
Find \(f + g\), \(f - g\), \(f g\), and \(\dfrac{f}{g}\) and their domains for \(f(x) = 2/(x + 1)\), \(g(x) = x/(x + 1)\).
(미작성)
정답
\((f+g)(x) = \dfrac{x + 2}{x + 1}\), \((f-g)(x) = \dfrac{2 - x}{x + 1}\), \((f g)(x) = 2x/(x + 1)^2\), \(\left(\dfrac{f}{g}\right)(x) = \dfrac{2}{x}\). Domain: \(x \neq -1\) (and \(x \neq 0\) for quotient).
위 정답과 비교하여 채점하세요:
해설
Both functions share the denominator \(x + 1\), so combinations simplify directly. Exclude \(x = -1\) throughout; for the quotient also exclude \(x = 0\) where \(g(x) = 0\).
11
Domain
오답
Find the domain of the function \(f(x) = \sqrt{x} + \sqrt{1 - x}\).
(미작성)
정답
\([0, 1]\)
위 정답과 비교하여 채점하세요:
해설
\(\sqrt{x}\) requires \(x \geq 0\), and \(\sqrt{1 - x}\) requires \(x \leq 1\). Intersecting these gives \(0 \leq x \leq 1\).
12
Domain
오답
Find the domain of the function \(g(x) = \sqrt{x + 1} - \dfrac{1}{x}\).
(미작성)
정답
\([-1, 0) \cup (0, \infty)\)
위 정답과 비교하여 채점하세요:
해설
\(\sqrt{x + 1}\) requires \(x \geq -1\), and \(\dfrac{1}{x}\) requires \(x \neq 0\). Combine: \(x \geq -1\) and \(x \neq 0\).
13
Domain
오답
Find the domain of the function \(h(x) = (x - 3)^{-\dfrac{1}{4}}\).
(미작성)
정답
\((3, \infty)\)
위 정답과 비교하여 채점하세요:
해설
\((x - 3)^{-\dfrac{1}{4}} = 1/\sqrt[4]{x - 3}\) requires \(x - 3 > 0\), i.e., \(x > 3\).
14
Domain
오답
Find the domain of the function \(k(x) = \dfrac{\sqrt{x + 3}}{x - 1}\).
(미작성)
정답
\([-3, 1) \cup (1, \infty)\)
위 정답과 비교하여 채점하세요:
해설
\(\sqrt{x + 3}\) requires \(x \geq -3\), and the denominator forces \(x \neq 1\).
15
Graphical addition
오답
Use graphical addition to sketch the graph of \(f + g\) from the figure.
(미작성)
정답
At each \(x\), plot the point with \(y\)-value equal to the sum of the heights of \(f\) and \(g\) at that \(x\).
위 정답과 비교하여 채점하세요:
해설
Graphical addition: for each \(x\), add the \(y\)-coordinates of the two graphs to obtain the corresponding point on \(f + g\).
16
Graphical addition
오답
Use graphical addition to sketch the graph of \(f + g\) from the figure.
(미작성)
정답
At each \(x\), plot the point with \(y\)-value equal to the sum of the heights of \(f\) and \(g\) at that \(x\).
위 정답과 비교하여 채점하세요:
해설
Apply graphical addition: at each chosen \(x\), add the \(y\)-coordinates from both curves to get \((f + g)(x)\).
17
Graph addition with formula
오답
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen to illustrate graphical addition for \(f(x) = \sqrt{1 + x}\), \(g(x) = \sqrt{1 - x}\).
(미작성)
정답
\(f\) has domain \([-1, \infty)\), \(g\) has domain \((-\infty, 1]\). \(f + g\) has domain \([-1, 1]\) and equals \(\sqrt{1 + x} + \sqrt{1 - x}\).
위 정답과 비교하여 채점하세요:
해설
Plot all three using a graphing utility on the common domain \([-1, 1]\). Verify \((f + g)(0) = 2\), the maximum value.
18
Graph addition with formula
오답
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen for \(f(x) = x^2\), \(g(x) = \sqrt{x}\).
(미작성)
정답
\((f + g)(x) = x^2 + \sqrt{x}\), domain \([0, \infty)\).
위 정답과 비교하여 채점하세요:
해설
Plot \(f(x) = x^2\) (parabola), \(g(x) = \sqrt{x}\) (square root), and their sum on the domain \(x \geq 0\).
19
Graph addition with formula
오답
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen for \(f(x) = x^2\), \(g(x) = \left(\dfrac{1}{3}\right) x^3\).
(미작성)
정답
\((f + g)(x) = x^2 + \left(\dfrac{1}{3}\right) x^3\), domain \(RR\).
위 정답과 비교하여 채점하세요:
해설
Plot \(f(x) = x^2\) (parabola), \(g(x) = x^3/3\) (cubic), and the sum \(x^2 + x^3/3\) together.
20
Graph addition with formula
오답
Draw the graphs of \(f\), \(g\), and \(f + g\) on a common screen for \(f(x) = \sqrt[4]{1 - x}\), \(g(x) = \sqrt{1 - x^2/9}\).
(미작성)
정답
\(f\) has domain \((-\infty, 1]\), \(g\) has domain \([-3, 3]\). \(f + g\) has domain \([-3, 1]\).
위 정답과 비교하여 채점하세요:
해설
Domains: \(f\) requires \(1 - x \geq 0\), i.e., \(x \leq 1\); \(g\) requires \(1 - x^2/9 \geq 0\), i.e., \(-3 \leq x \leq 3\). Intersection: \([-3, 1]\).
21
Composition evaluation
오답
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \(f(g(0))\), (b) \(g(f(0))\).
(미작성)
정답
(a) \(1\), (b) \(-23\)
위 정답과 비교하여 채점하세요:
해설
(a) \(g(0) = 2\), so \(f(g(0)) = f(2) = 6 - 5 = 1\). (b) \(f(0) = -5\), so \(g(f(0)) = g(-5) = 2 - 25 = -23\).
22
Composition evaluation
오답
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \(f(f(4))\), (b) \(g(g(3))\).
(미작성)
정답
(a) \(16\), (b) \(-47\)
위 정답과 비교하여 채점하세요:
해설
(a) \(f(4) = 7\), so \(f(f(4)) = f(7) = 21 - 5 = 16\). (b) \(g(3) = -7\), so \(g(g(3)) = g(-7) = 2 - 49 = -47\).
23
Composition evaluation
오답
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \((f \circ g)(-2)\), (b) \((g \circ f)(-2)\).
(미작성)
정답
(a) \(-11\), (b) \(-119\)
위 정답과 비교하여 채점하세요:
해설
(a) \(g(-2) = -2\), so \((f \circ g)(-2) = f(-2) = -6 - 5 = -11\). (b) \(f(-2) = -11\), so \((g \circ f)(-2) = g(-11) = 2 - 121 = -119\).
24
Composition evaluation
오답
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to evaluate (a) \((f \circ f)(-1)\), (b) \((g \circ g)(2)\).
(미작성)
정답
(a) \(-29\), (b) \(-2\)
위 정답과 비교하여 채점하세요:
해설
(a) \(f(-1) = -8\), so \((f \circ f)(-1) = f(-8) = -24 - 5 = -29\). (b) \(g(2) = -2\), so \((g \circ g)(2) = g(-2) = 2 - 4 = -2\).
25
Composition formula
오답
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to find (a) \((f \circ g)(x)\), (b) \((g \circ f)(x)\).
(미작성)
정답
(a) \(1 - 3x^2\), (b) \(-9x^2 + 30x - 23\)
위 정답과 비교하여 채점하세요:
해설
(a) \((f \circ g)(x) = f(2 - x^2) = 3(2 - x^2) - 5 = 1 - 3x^2\). (b) \((g \circ f)(x) = g(3x - 5) = 2 - (3x - 5)^2 = -9x^2 + 30x - 23\).
26
Composition formula
오답
Use \(f(x) = 3x - 5\) and \(g(x) = 2 - x^2\) to find (a) \((f \circ f)(x)\), (b) \((g \circ g)(x)\).
(미작성)
정답
(a) \(9x - 20\), (b) \(-x^4 + 4x^2 - 2\)
위 정답과 비교하여 채점하세요:
해설
(a) \((f \circ f)(x) = f(3x - 5) = 3(3x - 5) - 5 = 9x - 20\). (b) \((g \circ g)(x) = g(2 - x^2) = 2 - (2 - x^2)^2 = -x^4 + 4x^2 - 2\).
27
Composition from graphs
오답
Use the given graphs of \(f\) and \(g\) to evaluate \(f(g(2))\).
(미작성)
정답
Read \(g(2)\) from the graph of \(g\), then read \(f\) at that value from the graph of \(f\).
위 정답과 비교하여 채점하세요:
해설
First locate \(x = 2\) on the graph of \(g\) to obtain \(g(2)\). Then evaluate \(f\) at this output by reading the graph of \(f\).
28
Composition from graphs
오답
Use the given graphs of \(f\) and \(g\) to evaluate \(g(f(0))\).
(미작성)
정답
Read \(f(0)\) from the graph of \(f\), then read \(g\) at that value from the graph of \(g\).
위 정답과 비교하여 채점하세요:
해설
Find \(f(0)\) on the graph, then plug that result into \(g\) by reading the graph of \(g\).
29
Composition from graphs
오답
Use the given graphs of \(f\) and \(g\) to evaluate \((g \circ f)(4)\).
(미작성)
정답
Read \(f(4)\) from the graph of \(f\), then read \(g\) at that value from the graph of \(g\).
위 정답과 비교하여 채점하세요:
해설
\((g \circ f)(4) = g(f(4))\). First read \(f(4)\) from the graph; then evaluate \(g\) at that value.
30
Composition from graphs
오답
Use the given graphs of \(f\) and \(g\) to evaluate \((f \circ g)(0)\).
(미작성)
정답
Read \(g(0)\) from the graph of \(g\), then read \(f\) at that value from the graph of \(f\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(0) = f(g(0))\). Read \(g(0)\) from the graph; then evaluate \(f\) at that result.
31
Composition from graphs
오답
Use the given graphs of \(f\) and \(g\) to evaluate \((g \circ g)(-2)\).
(미작성)
정답
Read \(g(-2)\) from the graph of \(g\), then read \(g\) again at that value.
위 정답과 비교하여 채점하세요:
해설
\((g \circ g)(-2) = g(g(-2))\). Apply \(g\) twice using the graph: first to \(-2\), then to the resulting value.
32
Composition from graphs
오답
Use the given graphs of \(f\) and \(g\) to evaluate \((f \circ f)(4)\).
(미작성)
정답
Read \(f(4)\) from the graph of \(f\), then read \(f\) again at that value.
위 정답과 비교하여 채점하세요:
해설
\((f \circ f)(4) = f(f(4))\). Apply \(f\) twice using the graph: first to \(4\), then to the resulting value.
33
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = 2x + 3\), \(g(x) = 4x - 1\).
(미작성)
정답
\((f \circ g)(x) = 8x + 1\), \((g \circ f)(x) = 8x + 11\), \((f \circ f)(x) = 4x + 9\), \((g \circ g)(x) = 16x - 5\). All domains are \(RR\).
위 정답과 비교하여 채점하세요:
해설
Substitute and simplify: \((f \circ g)(x) = 2(4x - 1) + 3 = 8x + 1\); \((g \circ f)(x) = 4(2x + 3) - 1 = 8x + 11\); \((f \circ f)(x) = 2(2x + 3) + 3 = 4x + 9\); \((g \circ g)(x) = 4(4x - 1) - 1 = 16x - 5\).
34
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = 6x - 5\), \(g(x) = \dfrac{x}{2}\).
(미작성)
정답
\((f \circ g)(x) = 3x - 5\), \((g \circ f)(x) = 3x - \dfrac{5}{2}\), \((f \circ f)(x) = 36x - 35\), \((g \circ g)(x) = \dfrac{x}{4}\). All domains are \(RR\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = 6\left(\dfrac{x}{2}\right) - 5 = 3x - 5\); \((g \circ f)(x) = (6x - 5)/2 = 3x - \dfrac{5}{2}\); \((f \circ f)(x) = 6(6x - 5) - 5 = 36x - 35\); \((g \circ g)(x) = \left(\dfrac{x}{2}\right)/2 = \dfrac{x}{4}\).
35
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x^2\), \(g(x) = x + 1\).
(미작성)
정답
\((f \circ g)(x) = (x + 1)^2\), \((g \circ f)(x) = x^2 + 1\), \((f \circ f)(x) = x^4\), \((g \circ g)(x) = x + 2\). All domains are \(RR\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = (x + 1)^2\); \((g \circ f)(x) = x^2 + 1\); \((f \circ f)(x) = (x^2)^2 = x^4\); \((g \circ g)(x) = (x + 1) + 1 = x + 2\).
36
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x^3 + 2\), \(g(x) = \sqrt[3]{x}\).
(미작성)
정답
\((f \circ g)(x) = x + 2\), \((g \circ f)(x) = \sqrt[3]{x^3 + 2}\), \((f \circ f)(x) = (x^3 + 2)^3 + 2\), \((g \circ g)(x) = \sqrt[9]{x}\). All domains are \(RR\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = (\sqrt[3]{x})^3 + 2 = x + 2\); \((g \circ f)(x) = \sqrt[3]{x^3 + 2}\); \((f \circ f)(x) = (x^3 + 2)^3 + 2\); \((g \circ g)(x) = \sqrt[3]{\sqrt[3]{x}} = \sqrt[9]{x}\).
37
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = \dfrac{1}{x}\), \(g(x) = 2x + 4\).
(미작성)
정답
\((f \circ g)(x) = 1/(2x + 4)\), domain \(x \neq -2\). \((g \circ f)(x) = \dfrac{2}{x} + 4\), domain \(x \neq 0\). \((f \circ f)(x) = x\), domain \(x \neq 0\). \((g \circ g)(x) = 4x + 12\), domain \(RR\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = 1/(2x + 4)\); exclude \(x = -2\). \((g \circ f)(x) = 2\left(\dfrac{1}{x}\right) + 4 = \dfrac{2}{x} + 4\); exclude \(x = 0\). \((f \circ f)(x) = 1/\left(\dfrac{1}{x}\right) = x\) for \(x \neq 0\). \((g \circ g)(x) = 2(2x + 4) + 4 = 4x + 12\).
38
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x^2\), \(g(x) = \sqrt{x - 3}\).
(미작성)
정답
\((f \circ g)(x) = x - 3\), domain \(x \geq 3\). \((g \circ f)(x) = \sqrt{x^2 - 3}\), domain \(|x| \geq \sqrt{3}\). \((f \circ f)(x) = x^4\), domain \(RR\). \((g \circ g)(x) = \sqrt{\sqrt{x - 3} - 3}\), domain \(x \geq 12\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = (\sqrt{x - 3})^2 = x - 3\) for \(x \geq 3\). \((g \circ f)(x) = \sqrt{x^2 - 3}\) requires \(x^2 \geq 3\). \((f \circ f)(x) = x^4\). \((g \circ g)(x)\) requires \(\sqrt{x - 3} \geq 3\), i.e., \(x \geq 12\).
39
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = |x|\), \(g(x) = 2x + 3\).
(미작성)
정답
\((f \circ g)(x) = |2x + 3|\), \((g \circ f)(x) = 2 |x| + 3\), \((f \circ f)(x) = |x|\), \((g \circ g)(x) = 4x + 9\). All domains are \(RR\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = |2x + 3|\); \((g \circ f)(x) = 2 |x| + 3\); \((f \circ f)(x) = |abs(x)| = |x|\); \((g \circ g)(x) = 2(2x + 3) + 3 = 4x + 9\).
40
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x - 4\), \(g(x) = |x + 4|\).
(미작성)
정답
\((f \circ g)(x) = |x + 4| - 4\), \((g \circ f)(x) = |x|\), \((f \circ f)(x) = x - 8\), \((g \circ g)(x) = |abs(x + 4) + 4| = |x + 4| + 4\). All domains are \(RR\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = |x + 4| - 4\); \((g \circ f)(x) = |(x - 4) + 4| = |x|\); \((f \circ f)(x) = (x - 4) - 4 = x - 8\); \((g \circ g)(x) = |abs(x + 4) + 4|\), and since \(|x + 4| + 4 \geq 0\), this equals \(|x + 4| + 4\).
41
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x/(x + 1)\), \(g(x) = 2x - 1\).
(미작성)
정답
\((f \circ g)(x) = \dfrac{2x - 1}{2x}\), domain \(x \neq 0\). \((g \circ f)(x) = \dfrac{x - 1}{x + 1}\), domain \(x \neq -1\). \((f \circ f)(x) = x/(2x + 1)\), domain \(x \neq -1, -\dfrac{1}{2}\). \((g \circ g)(x) = 4x - 3\), domain \(RR\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = \dfrac{2x - 1}{(2x - 1) + 1} = \dfrac{2x - 1}{2x}\). \((g \circ f)(x) = 2(x/(x+1)) - 1 = \dfrac{x - 1}{x + 1}\). \((f \circ f)(x) = \dfrac{x/(x+1)}{(x/(x+1)) + 1} = x/(2x + 1)\). \((g \circ g)(x) = 2(2x - 1) - 1 = 4x - 3\).
42
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = \dfrac{1}{\sqrt{x}}\), \(g(x) = x^2 - 4x\).
(미작성)
정답
\((f \circ g)(x) = \dfrac{1}{\sqrt{x^2 - 4x}}\), domain \(x < 0\) or \(x > 4\). \((g \circ f)(x) = \dfrac{1}{x} - \dfrac{4}{\sqrt{x}}\), domain \(x > 0\). \((f \circ f)(x) = \sqrt[4]{x}\), domain \(x > 0\). \((g \circ g)(x) = (x^2 - 4x)^2 - 4(x^2 - 4x)\), domain \(RR\).
위 정답과 비교하여 채점하세요:
해설
\(f\) requires \(x > 0\) and \(g\) has domain \(RR\). \((f \circ g)\) requires \(g(x) > 0\), i.e., \(x(x - 4) > 0\), giving \(x < 0\) or \(x > 4\). \((g \circ f)\) requires \(x > 0\). \((f \circ f)(x) = 1/\sqrt{\dfrac{1}{\sqrt{x}}} = x^{\dfrac{1}{4}}\) for \(x > 0\).
43
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = x/(x + 1)\), \(g(x) = \dfrac{1}{x}\).
(미작성)
정답
\((f \circ g)(x) = 1/(1 + x)\), domain \(x \neq 0, -1\). \((g \circ f)(x) = (x + 1)/x\), domain \(x \neq 0, -1\). \((f \circ f)(x) = x/(2x + 1)\), domain \(x \neq -1, -\dfrac{1}{2}\). \((g \circ g)(x) = x\), domain \(x \neq 0\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = \dfrac{\dfrac{1}{x}}{\left(\dfrac{1}{x}\right) + 1} = 1/(1 + x)\). \((g \circ f)(x) = 1/(x/(x+1)) = (x + 1)/x\). \((f \circ f)(x) = \dfrac{x/(x+1)}{(x/(x+1)) + 1} = x/(2x + 1)\). \((g \circ g)(x) = 1/\left(\dfrac{1}{x}\right) = x\).
44
Find compositions
오답
Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\), and their domains, for \(f(x) = \dfrac{2}{x}\), \(g(x) = x/(x + 2)\).
(미작성)
정답
\((f \circ g)(x) = (2(x + 2))/x = 2 + \dfrac{4}{x}\), domain \(x \neq 0, -2\). \((g \circ f)(x) = 2/(2 + 2x) = 1/(1 + x)\), domain \(x \neq 0, -1\). \((f \circ f)(x) = x\), domain \(x \neq 0\). \((g \circ g)(x) = x/(3x + 4)\), domain \(x \neq -2, -\dfrac{4}{3}\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = 2/(x/(x + 2)) = 2(x + 2)/x\). \((g \circ f)(x) = \dfrac{\dfrac{2}{x}}{\left(\dfrac{2}{x}\right) + 2} = 2/(2 + 2x) = 1/(1 + x)\). \((f \circ f)(x) = 2/\left(\dfrac{2}{x}\right) = x\). \((g \circ g)(x) = \dfrac{x/(x+2)}{(x/(x+2)) + 2} = x/(3x + 4)\).
45
Triple composition
오답
Find \(f \circ g \circ h\) for \(f(x) = x - 1\), \(g(x) = \sqrt{x}\), \(h(x) = x - 1\).
(미작성)
정답
\((f \circ g \circ h)(x) = \sqrt{x - 1} - 1\)
위 정답과 비교하여 채점하세요:
해설
\(h(x) = x - 1\), then \(g(h(x)) = \sqrt{x - 1}\), then \(f(g(h(x))) = \sqrt{x - 1} - 1\). Domain: \(x \geq 1\).
46
Triple composition
오답
Find \(f \circ g \circ h\) for \(f(x) = \dfrac{1}{x}\), \(g(x) = x^3\), \(h(x) = x^2 + 2\).
(미작성)
정답
\((f \circ g \circ h)(x) = 1/(x^2 + 2)^3\)
위 정답과 비교하여 채점하세요:
해설
\(h(x) = x^2 + 2\), then \(g(h(x)) = (x^2 + 2)^3\), then \(f(g(h(x))) = 1/(x^2 + 2)^3\). Domain: \(RR\).
47
Triple composition
오답
Find \(f \circ g \circ h\) for \(f(x) = x^4 + 1\), \(g(x) = x - 5\), \(h(x) = \sqrt{x}\).
(미작성)
정답
\((f \circ g \circ h)(x) = (\sqrt{x} - 5)^4 + 1\)
위 정답과 비교하여 채점하세요:
해설
\(h(x) = \sqrt{x}\), then \(g(h(x)) = \sqrt{x} - 5\), then \(f(g(h(x))) = (\sqrt{x} - 5)^4 + 1\). Domain: \(x \geq 0\).
48
Triple composition
오답
Find \(f \circ g \circ h\) for \(f(x) = \sqrt{x}\), \(g(x) = x/(x - 1)\), \(h(x) = \sqrt[3]{x}\).
(미작성)
정답
\((f \circ g \circ h)(x) = \sqrt{\sqrt[3]{x}/(\sqrt[3]{x} - 1)}\)
위 정답과 비교하여 채점하세요:
해설
\(h(x) = \sqrt[3]{x}\), then \(g(h(x)) = \sqrt[3]{x}/(\sqrt[3]{x} - 1)\), then \(f(g(h(x))) = \sqrt{\sqrt[3]{x}/(\sqrt[3]{x} - 1)}\). Domain requires the radicand non-negative and \(\sqrt[3]{x} \neq 1\).
49
Express as composition
오답
Express the function \(F(x) = (x - 9)^5\) in the form \(f \circ g\).
(미작성)
정답
\(g(x) = x - 9\), \(f(x) = x^5\)
위 정답과 비교하여 채점하세요:
해설
Let the inner function \(g(x) = x - 9\) and the outer \(f(x) = x^5\). Then \(f(g(x)) = (x - 9)^5 = F(x)\).
50
Express as composition
오답
Express the function \(F(x) = \sqrt{x} + 1\) in the form \(f \circ g\).
(미작성)
정답
\(g(x) = \sqrt{x}\), \(f(x) = x + 1\)
위 정답과 비교하여 채점하세요:
해설
Let \(g(x) = \sqrt{x}\) and \(f(x) = x + 1\). Then \(f(g(x)) = \sqrt{x} + 1 = F(x)\).
51
Express as composition
오답
Express the function \(G(x) = x^2/(x^2 + 4)\) in the form \(f \circ g\).
(미작성)
정답
\(g(x) = x^2\), \(f(x) = x/(x + 4)\)
위 정답과 비교하여 채점하세요:
해설
Let \(g(x) = x^2\) and \(f(x) = x/(x + 4)\). Then \(f(g(x)) = x^2/(x^2 + 4) = G(x)\).
52
Express as composition
오답
Express the function \(G(x) = 1/(x + 3)\) in the form \(f \circ g\).
(미작성)
정답
\(g(x) = x + 3\), \(f(x) = \dfrac{1}{x}\)
위 정답과 비교하여 채점하세요:
해설
Let \(g(x) = x + 3\) and \(f(x) = \dfrac{1}{x}\). Then \(f(g(x)) = 1/(x + 3) = G(x)\).
53
Express as composition
오답
Express the function \(H(x) = |1 - x^3|\) in the form \(f \circ g\).
(미작성)
정답
\(g(x) = 1 - x^3\), \(f(x) = |x|\)
위 정답과 비교하여 채점하세요:
해설
Let \(g(x) = 1 - x^3\) and \(f(x) = |x|\). Then \(f(g(x)) = |1 - x^3| = H(x)\).
54
Express as composition
오답
Express the function \(H(x) = \sqrt{1 + \sqrt{x}}\) in the form \(f \circ g\).
(미작성)
정답
\(g(x) = 1 + \sqrt{x}\), \(f(x) = \sqrt{x}\)
위 정답과 비교하여 채점하세요:
해설
Let \(g(x) = 1 + \sqrt{x}\) and \(f(x) = \sqrt{x}\). Then \(f(g(x)) = \sqrt{1 + \sqrt{x}} = H(x)\).
55
Express as triple composition
오답
Express the function \(F(x) = 1/(x^2 + 1)\) in the form \(f \circ g \circ h\).
(미작성)
정답
\(h(x) = x^2\), \(g(x) = x + 1\), \(f(x) = \dfrac{1}{x}\)
위 정답과 비교하여 채점하세요:
해설
Let \(h(x) = x^2\), \(g(x) = x + 1\), \(f(x) = \dfrac{1}{x}\). Then \(f(g(h(x))) = 1/(x^2 + 1) = F(x)\).
56
Express as triple composition
오답
Express the function \(F(x) = \sqrt[3]{\sqrt{x} - 1}\) in the form \(f \circ g \circ h\).
(미작성)
정답
\(h(x) = \sqrt{x}\), \(g(x) = x - 1\), \(f(x) = \sqrt[3]{x}\)
위 정답과 비교하여 채점하세요:
해설
Let \(h(x) = \sqrt{x}\), \(g(x) = x - 1\), \(f(x) = \sqrt[3]{x}\). Then \(f(g(h(x))) = \sqrt[3]{\sqrt{x} - 1} = F(x)\).
57
Express as triple composition
오답
Express the function \(G(x) = (4 + \sqrt[3]{x})^9\) in the form \(f \circ g \circ h\).
(미작성)
정답
\(h(x) = \sqrt[3]{x}\), \(g(x) = 4 + x\), \(f(x) = x^9\)
위 정답과 비교하여 채점하세요:
해설
Let \(h(x) = \sqrt[3]{x}\), \(g(x) = 4 + x\), \(f(x) = x^9\). Then \(f(g(h(x))) = (4 + \sqrt[3]{x})^9 = G(x)\).
58
Express as triple composition
오답
Express the function \(G(x) = 2/(3 + \sqrt{x})^2\) in the form \(f \circ g \circ h\).
(미작성)
정답
\(h(x) = \sqrt{x}\), \(g(x) = (3 + x)^2\), \(f(x) = \dfrac{2}{x}\)
위 정답과 비교하여 채점하세요:
해설
Let \(h(x) = \sqrt{x}\), \(g(x) = (3 + x)^2\), \(f(x) = \dfrac{2}{x}\). Then \(f(g(h(x))) = 2/(3 + \sqrt{x})^2 = G(x)\).
59
Applications - Revenue
오답
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 \(x\).
(미작성)
정답
\(R(x) = (0.15 - 0.000002 x) \cdot x = 0.15 x - 0.000002 x^2\)
위 정답과 비교하여 채점하세요:
해설
Revenue equals price per sticker times number of stickers: \(R(x) = (0.15 - 0.000002 x)(x)\).
60
Applications - Profit
오답
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.)
(미작성)
정답
\(P(x) = R(x) - C(x) = (0.15 x - 0.000002 x^2) - (0.095 x - 0.0000005 x^2) = 0.055 x - 0.0000015 x^2\)
위 정답과 비교하여 채점하세요:
해설
Subtract cost from revenue: \(P(x) = (0.15 - 0.000002 x) x - (0.095 x - 0.0000005 x^2) = 0.055 x - 0.0000015 x^2\).
61
Applications - Area of a Ripple
오답
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?
(미작성)
정답
(a) \(g(t) = 60 t\), (b) \(f(r) = \pi r^2\), (c) \((f \circ g)(t) = 3600 \pi t^2\), the area of the ripple at time \(t\) (cm\(^2\)).
위 정답과 비교하여 채점하세요:
해설
Radius grows at \(60\) cm/s, so \(g(t) = 60 t\). Circle area: \(f(r) = \pi r^2\). Composition: \(f(g(t)) = \pi (60 t)^2 = 3600 \pi t^2\).
62
Applications - Inflating a Balloon
오답
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?
(미작성)
정답
(a) \(f(t) = t\), (b) \(g(r) = \left(\dfrac{4}{3}\right) \pi r^3\), (c) \((g \circ f)(t) = \left(\dfrac{4}{3}\right) \pi t^3\), the volume of the balloon at time \(t\) (cm\(^3\)).
위 정답과 비교하여 채점하세요:
해설
Radius increases at \(1\) cm/s, so \(f(t) = t\). Sphere volume: \(g(r) = \left(\dfrac{4}{3}\right) \pi r^3\). Composition: \(g(f(t)) = \left(\dfrac{4}{3}\right) \pi t^3\).
63
Applications - Area of a Balloon
오답
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).
(미작성)
정답
\(S(t) = 16 \pi t^2\)
위 정답과 비교하여 채점하세요:
해설
Radius: \(r(t) = 2 t\). Sphere surface area: \(S = 4 \pi r^2 = 4 \pi (2 t)^2 = 16 \pi t^2\).
64
Applications - Multiple Discounts
오답
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?
(미작성)
정답
(a) \(f(x) = 0.8 x\), (b) \(g(x) = x - 50\), (c) \((f \circ g)(x) = 0.8(x - 50) = 0.8 x - 40\), \((g \circ f)(x) = 0.8 x - 50\). \((g \circ f)(x)\) gives the lower price.
위 정답과 비교하여 채점하세요:
해설
Comparing the two: \((g \circ f)(x) - (f \circ g)(x) = -10\), so \((g \circ f)\) is \(\$10\) cheaper. Apply the discount first, then the coupon.
65
Applications - Multiple Discounts
오답
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?
(미작성)
정답
(a) \(f(x) = 0.9 x\), (b) \(g(x) = x - 100\), (c) \((f \circ g)(x) = 0.9 x - 90\), \((g \circ f)(x) = 0.9 x - 100\). \((g \circ f)\) is the better deal.
위 정답과 비교하여 채점하세요:
해설
\((g \circ f)(x) - (f \circ g)(x) = -10\), so applying the \(10%\) discount first then the rebate saves \(\$10\) more than the reverse order.
66
Applications - Airplane Trajectory
오답
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 \(t\).
(미작성)
정답
(a) \(s(d) = \sqrt{d^2 + 1}\), (b) \(d(t) = 350 t\), (c) \(s(t) = \sqrt{122500 t^2 + 1}\)
위 정답과 비교하여 채점하세요:
해설
By the Pythagorean theorem with vertical leg \(1\) and horizontal leg \(d\): \(s = \sqrt{d^2 + 1}\). Distance traveled: \(d = 350 t\). Composition: \(s(t) = \sqrt{(350 t)^2 + 1} = \sqrt{122500 t^2 + 1}\).
67
Discovery - Compound Interest
오답
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 \(A\).
(미작성)
정답
\((A \circ A)(x) = 1.05^2 x\), \((A \circ A \circ A)(x) = 1.05^3 x\), \((A \circ A \circ A \circ A)(x) = 1.05^4 x\). The \(n\)-fold composition: \(A^n(x) = 1.05^n x\), the value after \(n\) years.
위 정답과 비교하여 채점하세요:
해설
Each composition multiplies by \(1.05\), so the \(n\)-fold composition multiplies by \(1.05^n\). This is the amount in the account after \(n\) years of compounding.
68
Discovery - Composing Linear Functions
오답
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?
(미작성)
정답
Yes, \(f \circ g\) is a line with slope \(m_1 m_2\).
위 정답과 비교하여 채점하세요:
해설
\((f \circ g)(x) = m_1(m_2 x + b_2) + b_1 = m_1 m_2 x + (m_1 b_2 + b_1)\). This is linear with slope equal to the product of the two slopes.
69
Discovery - Solving for Unknown Function
오답
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 \(f \circ g = h\).
(미작성)
정답
First: \(f(x) = x^2 + 6\). Second: \(g(x) = x^2 + x - 1\).
위 정답과 비교하여 채점하세요:
해설
First part: \(h(x) = 4x^2 + 4x + 7 = (2x + 1)^2 + 6 = (g(x))^2 + 6\), so \(f(x) = x^2 + 6\). Second part: solve \(3 g(x) + 5 = 3 x^2 + 3 x + 2\), giving \(g(x) = (3 x^2 + 3 x - 3)/3 = x^2 + x - 1\).
70
Discovery - Odd and Even Compositions
오답
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?
(미작성)
정답
(a) Yes, \(h\) is even. (b) Not necessarily. (c) \(h\) is odd. (d) \(h\) is even.
위 정답과 비교하여 채점하세요:
해설
(a) If \(g\) even: \(h(-x) = f(g(-x)) = f(g(x)) = h(x)\), even. (b) Just \(g\) odd: \(h(-x) = f(-g(x))\), no general property. (c) Both odd: \(h(-x) = f(-g(x)) = -f(g(x)) = -h(x)\), odd. (d) \(g\) odd, \(f\) even: \(h(-x) = f(-g(x)) = f(g(x)) = h(x)\), even.
다시 풀기
점수 변화 (최근 2회)
0%
7/23
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7/28
| # | 날짜 | 점수 | 정답률 | |
|---|---|---|---|---|
| 1 | 2026-07-28 11:42 | 0 / 70 | 0% | 보기 |
| 현재 | 2026-07-23 20:35 | 0 / 70 | 0% |