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Checkpoint: Assess Your Understanding, pages 287–290
4.1
1. Multiple Choice Given the graphs of y ϭ f(x) and y ϭ g(x), which y
f(x)
graph below represents y ϭ ? 2 y ϭ f(x)
g(x)
x
A. y B. y Ϫ2 0 2
2 2 Ϫ2 y ϭ g(x)
x x
Ϫ2 0 2 0 2
Ϫ2 Ϫ2
C. y D. y
2 2
x x
0 2 Ϫ2 0 2
Ϫ2
12 Chapter 4: Combining Functions—Checkpoint—Solutions DO NOT COPY. ©P

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2. Use the graphs of y ϭ f(x) and y ϭ g(x) to sketch the graph of each
given function. Identify its domain and range; approximate the
range where necessary.
a) y ϭ f(x) ϩ g(x) b) y ϭ f(x) Ϫ g(x)
y ϭ f(x) y y
4 4 y ϭ f(x)
y ϭ f(x)ϩg(x)
2 2
x y ϭ f(x) Ϫ g(x) x
Ϫ4 Ϫ2 0 Ϫ4 Ϫ2 0
y ϭ g(x) Ϫ2 y ϭ g(x) Ϫ2
From the graphs:
f(x)
x f(x) g(x) f(x) ؉ g(x) f(x) ؊ g(x) f(x) # g(x)
g(x)
؊4 5 0 5 5 0 undefined
؊3 2 ؊1 1 3 ؊2 ؊2
؊2 1 Џ ؊1.4 Џ ؊0.4 Џ 2.4 Џ ؊1.4 Џ ؊ 0.7
؊1 2 Џ ؊1.7 Џ 0.3 Џ 3.7 Џ ؊3.5 Џ ؊1.2
0 5 ؊2 3 7 ؊10 ؊2.5
Plot points at: (؊4, 5), Plot points at: (؊4, 5), (؊3, 3),
(؊3, 1), (؊2, ؊0.4), (؊2, 2.4), (؊1, 3.7)
(؊1, 0.3), (0, 3) Join the points with a smooth curve.
Join the points with a Domain: x » ؊4
smooth curve. Approximate range: y » 2.4
Domain: x » ؊4
Approximate range:
y » ؊0.4
©P DO NOT COPY. Chapter 4: Combining Functions—Checkpoint—Solutions 13

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f(x)
c) y = f(x) # g(x) d) y =
g(x)
y ϭ f(x) y y ϭ f(x ) y
4 4
2 2
x x
Ϫ2 0 Ϫ4 0
y ϭ g(x) y ϭ g(x)
Ϫ2 Ϫ2
y ϭ f(x )·g(x) f(x)
yϭ
g(x)
Plot points at: (؊4, 0), Plot points at: (؊3, ؊2), (؊2, ؊0.7),
(؊3, ؊2), (؊2, ؊1.4), (؊1, ؊1.2), (0, ؊2.5)
(؊1, ؊3.5) Since g(؊4) ‫ ,0 ؍‬draw an asymptote
Join the points with a smooth at x ‫.4؊ ؍‬
curve. Join the points with a smooth curve.
Domain: x » ؊4 Domain: x>؊4
Range: y ◊ 0 Approximate range: y ◊ ؊0.7
4.2
√
3. Multiple Choice Given f(x) = x - 2 and g(x) = x, what is the
domain of h(x) = f(x) # g(x)?
A. x H ‫ޒ‬ B. x Z 2 C. x > 2 D. x ≥ 0
4. Use f(x) = x2 + x - 20.
a) Write explicit equations for two functions g(x) and k(x) so that
f(x) = g(x) # k(x).
Sample response:
Factor: f(x) ‫( ؍‬x ؉ 5)(x ؊ 4)
So, g(x) ‫ ؍‬x ؉ 5 and k(x) ‫ ؍‬x ؊ 4
b) Write explicit equations for three functions g(x), h(x), and k(x) so
that f(x) = g(x) - h(x) - k(x).
Sample response:
f(x) ‫ ؍‬x2 ؉ x ؊ 20
f(x) ‫ ؍‬x2 ؊ (؊x) ؊ 20
So, g(x) ‫ ؍‬x2; h(x) ‫؊ ؍‬x; and k(x) ‫02 ؍‬
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c) Write explicit equations for two functions g(x) and k(x) so that
g(x)
f(x) = .
k(x)
Sample response:
Multiply and divide x2 ؉ x ؊ 20 by a non-zero expression.
(x2 ؉ x ؊ 20)(x2 ؉ 4)
f(x) ‫؍‬
x2 ؉ 4
So, g(x) ‫( ؍‬x ؉ x ؊ 20)(x2 ؉ 4) and k(x) ‫ ؍‬x2 ؉ 4
2
1 √
5. Use f(x) = 3x2 - 1, g(x) = , and h(x) = x - 5.
x + 2
i) Write an explicit equation for each function below.
ii) State the domain and range of each function; approximate the
range where necessary.
a) h(x) = f(x) + g(x) b) d(x) = g(x) - h(x)
1 1 √
i) h(x) ‫3 ؍‬x2 ؊ 1 ؉ i) d(x) ‫؍‬ ؊ x؊5
x؉2 x؉2
ii) The domain is: x ؊2 ii) The domain is: x » 5
Use technology; the range Use technology; the range is:
is: y ç ‫ޒ‬ y ◊ 1
7
h(x)
c) p(x) = f(x) # g(x) d) q(x) =
g(x)
√
x؊5
b
1
i) p(x) ‫3( ؍‬x ؊ 1)a
2
i) q(x) ‫؍‬
x؉2 1
x؉2
3x2 ؊ 1 √
p(x) ‫؍‬ q(x) ‫( ؍‬x ؉ 2) x ؊ 5
x؉2
ii) The domain is: x ؊2 ii) The domain is: x » 5
Use technology; the range is Use technology; the range is:
approximately: y » ؊0.5 or y » 0
y ◊ ؊23.5
©P DO NOT COPY. Chapter 4: Combining Functions—Checkpoint—Solutions 15