9.5 Composition of Functions
Chapter
Chapter 9
Section
9.5
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Solutions 58 Videos

Use f(x)=2x-3 and g(x) = 1 -x^2 to evaluate the following expressions.

  • f(g(0))
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Q1a

Use f(x)=3x-3 and g(x) = 1 -x^2 to evaluate the following expressions.

  • g(f(4))
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Q1b

Use f(x)=3x-3 and g(x) = 1 -x^2 to evaluate the following expressions.

  • (f\circ g)(-8)
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Q1c

Use f(x)=3x-3 and g(x) = 1 -x^2 to evaluate the following expressions.

  • \displaystyle{(g\circ g)\left(\frac{1}{2}\right)}
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Q1d

Use f(x)=3x-3 and g(x) = 1 -x^2 to evaluate the following expressions.

  • (f\circ f^{-1})(1)
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Q1e

Use f(x)=3x-3 and g(x) = 1 -x^2 to evaluate the following expressions.

  • (g\circ g)(2)
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Q1f

Given f=\{(0,1),(1,2),(2,5),(3,10)\} and g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}, determine the following values.

  • (g\circ f)(2)
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Q2a

Given f=\{(0,1),(1,2),(2,5),(3,10)\} and g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}, determine the following values.

  • (f\circ f)(1)
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Q2b

Given f=\{(0,1),(1,2),(2,5),(3,10)\} and g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}, determine the following values.

  • (f\circ g)(5)
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Q2c

Given f=\{(0,1),(1,2),(2,5),(3,10)\} and g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}, determine the following values.

  • (f\circ g)(-2)
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Q2d

Given f=\{(0,1),(1,2),(2,5),(3,10)\} and g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}, determine the following values.

  • (f\circ f^{-1})(2)
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Q2e

Given f=\{(0,1),(1,2),(2,5),(3,10)\} and g=\{(2,0),(3,1),(4,2),(5,3),(6,4)\}, determine the following values.

  • (g^{-1}\circ f)(1)
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Q2f

Use the graph of f and g to evaluate each expression.

  • f(g(2))
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Q3a

Use the graph of f and g to evaluate each expression.

  • g(f(4))
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Q3b

Use the graph of f and g to evaluate each expression.

  • (g\circ g)(-2)
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Q3c

Use the graph of f and g to evaluate each expression.

  • (f\circ f)(2)
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Q3d

For a car travelling at a constant speed of 80 km h, the distance driven, d kilometres, is represented by d(t)=80t, where t is the time in hours. The cost of gasoline, in dollars, for the drive is represented by C(d)=0.09d

Determine C(d(5)) numerically, and interpret your results.

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Q4

In each case, functions f and g are defined for x\in\mathbb{R}. For each pair of functions, determine the expression and the domain of f(g(x)) and g(f(x)).

f(x)=3x^2,g(x)=x-1

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Q5a

In each case, functions f and g are defined for x\in\mathbb{R}. For each pair of functions, determine the expression and the domain of f(g(x)) and g(f(x)). Graph each result.

  • f(x)=2x^2+x,g(x)=x^2+1
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Q5b

In each case, functions f and g are defined for x\in\mathbb{R}. For each pair of functions, determine the expression and the domain of f(g(x)) and g(f(x)). Graph each result.

*2x^3-3x^2+x-1,g(x)=2x-1

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1.08mins
Q5c

In each case, functions f and g are defined for x\in\mathbb{R}. For each pair of functions, determine the expression and the domain of f(g(x)) and g(f(x)). Graph each result.

  • f(x)=x^4-x^2,g(x)=x+1
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Q5d

In each case, functions f and g are defined for x\in\mathbb{R}. For each pair of functions, determine the expression and the domain of f(g(x)) and g(f(x)). Graph each result.

  • f(x)=\sin x,g(x)=4x
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Q5e

In each case, functions f and g are defined for x\in\mathbb{R}. For each pair of functions, determine the expression and the domain of f(g(x)) and g(f(x)). Graph each result.

f(x)=|x|,g(x)=x+5

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Q5f

f(x)=3x,g(x)=\sqrt{x-4}

  • determine the defining equation for f\circ g and g\circ f

  • determine the domain and range of f\circ g and g\circ f

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Q6a

For each of the following,

  • determine the defining equation for f\circ g and g\circ f

  • determine the domain and range of f\circ g and g\circ f

  • f(x)=\sqrt{x},g(x)=3x+1

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Q6b

For each of the following,

  • determine the defining equation for f\circ g and g\circ f

  • determine the domain and range of f\circ g and g\circ f

  • f(x)=\sqrt{4-x^2},g(x)=x^2

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Q6c

f(x)=2^x,g(x)=\sqrt{x-1}

  • determine the defining equation for f\circ g and g\circ f

  • determine the domain and range of f\circ g and g\circ f

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Q6d

f(x)=10^x,g(x)=\log x

  • determine the defining equation for f\circ g and g\circ f

  • determine the domain and range of f\circ g and g\circ f

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Q6e

f(x)=\sin x,g(x)=5^{2x}+1

  • determine the defining equation for f\circ g and g\circ f

  • determine the domain and range of f\circ g and g\circ f

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Q6f

For each function h, find two functions, f g, such that h(x)=f(g(x)).

  • h(x)=\sqrt{x^2+6}
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Q7a

For each function h, find two functions, f g, such that h(x)=f(g(x)).

  • h(x)=(5x-8)^6
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0.36mins
Q7b

For each function h, find two functions, f g, such that h(x)=f(g(x)).

  • h(x)=2^{(6x+7)}
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Q7c

For each function h, find two functions, f g, such that h(x)=f(g(x)).

  • \displaystyle{h(x)=\frac{1}{x^3-7x+2}}
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Q7d

For each function h, find two functions, f g, such that h(x)=f(g(x)).

  • h(x)=\sin^2(10x+5)
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1.09mins
Q7e

For each function h, find two functions, f g, such that h(x)=f(g(x)).

  • h(x)=\sqrt[3]{(x+4)^2}
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0.39mins
Q7f

Let f(x)=2x-1 and g(x)=x^2. Determine (f\circ g)(x).

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0.19mins
Q8a

Let f(x)=2x-1 and g(x)=x^2.

Graph f,g, and f\circ g on the same set of axes

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Q8b

Let f(x)=2x-1 and g(x)=x^2

Describe the graph of f\circ g as a transformation of the graph of y=g(x).

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0.42mins
Q8c

Let f(x)=2x-1 and g(x)=3x+2.

  • Determine f(g(x)), and describe its graph as a transformation of g(x).
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1.44mins
Q9a

Let f(x)=2x-1 and g(x)=3x+2.

  • Determine g(f(x)), and describe its graph as a transformation of f(x)
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Q9b

A banquet hall charges $975 to rent a reception room, plus $39.95 per person. Next month, however, the banquet hall will be offering a 20% discount off the total bill. Express this discounted cost as a function of the number of people attending.

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Q10

The function f(x)=0.08x represents the sales tax owed on a purchase with a selling price of x dollars, and the function g(x)=0.75x represents the sale price of an item with a price tag of x dollars during a 25% off sale. Write a function that represents the sales tax owed on an item with a price tag of x dollars during a 25% off sale.

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Q11

An airplane passes directly over a radar station at time t=0. The plane maintains an altitude of 4 km and is flying at a speed of 560 km/h. Let d represent the distance from the radar station to the plane, and let s represent the horizontal distance travelled by the plane since it passed over the radar station.

a) Express d as a function of s, and s as a function of t.

b) Use composition to express the distance between the plane and the radar station as a function of time.

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1.36mins
Q12

In a vehicle test lab, the speed of a car, v kilometres per hour, at a time of t hours is represented by v(t)=40+3t+t^2. The rate of gasoline consumption of the car, c litres per kilometre, at a speed of v kilometres per hour is represented by \displaystyle{c(v)=\left(\frac{v}{500}-0.1\right)^2+0.15}. Determine algebraically c(v(t)), the rate of gasoline consumption as a function of time. Determine, using technology, the time when the car is running most economically during a 4 h simulation.

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Q13

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(f\circ g)(x)
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1.05mins
Q14a

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(f\circ h)(x)
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0.39mins
Q14b

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(f\circ k)(x)
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0.39mins
Q14c

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(f\circ m)(x)
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0.37mins
Q14d

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(f\circ n)(x)
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1.11mins
Q14e

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(f\circ p)(x)
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0.49mins
Q14f

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(g\circ f)(x)
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Q14g

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

*y=(h\circ f)(x)

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Q14h

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

g(x)=x+3

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Q14i

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

Which graph is y=(m\circ f)(x)?

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Q14j

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(n\circ f)(x)
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Q14k

Given the graph of y=f(x) shown and the functions below, match the correct composition with each graph. Justify your choices.

i) g(x)=x+3

ii) m(x)=2x

iii) h(x)=x-3

iv) n(x)=-0.5x

v) k(x)=-x

vi) p(x)=x-4

  • y=(p\circ f)(x)
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Q14l

If y=3x-2,x=3t+2, and t=3k-2, find an expression for y=f(k).

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Q16a

Express y as a function of k if y=2x+5,x=\sqrt{3t-1}, and t=3k-5.

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Q16b
Lectures 8 Videos

ex. Decompose \displaystyle y = \frac{1}{\sqrt{1 -x^2}}

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Decomposition example

ex. Find f, g, h for f\circ g\circ h(x) = \sqrt{x^2 - 1}

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Decomposition example 3