Use f(x)=2x-3
and g(x) = 1 -x^2
to evaluate the following expressions.
f(g(0))
Use f(x)=3x-3
and g(x) = 1 -x^2
to evaluate the following expressions.
g(f(4))
Use f(x)=3x-3
and g(x) = 1 -x^2
to evaluate the following expressions.
(f\circ g)(-8)
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)}
Use f(x)=3x-3
and g(x) = 1 -x^2
to evaluate the following expressions.
(f\circ f^{-1})(1)
Use f(x)=3x-3
and g(x) = 1 -x^2
to evaluate the following expressions.
(g\circ g)(2)
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)
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)
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)
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)
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)
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)
Use the graph of f
and g
to evaluate each expression.
f(g(2))
Use the graph of f
and g
to evaluate each expression.
g(f(4))
Use the graph of f
and g
to evaluate each expression.
(g\circ g)(-2)
Use the graph of f
and g
to evaluate each expression.
(f\circ f)(2)
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.
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
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
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
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
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
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
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
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
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
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
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
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
For each function h
, find two functions, f
g
, such that h(x)=f(g(x))
.
h(x)=\sqrt{x^2+6}
For each function h
, find two functions, f
g
, such that h(x)=f(g(x))
.
h(x)=(5x-8)^6
For each function h
, find two functions, f
g
, such that h(x)=f(g(x))
.
h(x)=2^{(6x+7)}
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}}
For each function h
, find two functions, f
g
, such that h(x)=f(g(x))
.
h(x)=\sin^2(10x+5)
For each function h
, find two functions, f
g
, such that h(x)=f(g(x))
.
h(x)=\sqrt[3]{(x+4)^2}
Let f(x)=2x-1
and g(x)=x^2
. Determine (f\circ g)(x)
.
Let f(x)=2x-1
and g(x)=x^2
.
Graph f,g
, and f\circ g
on the same set of axes
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)
.
Let f(x)=2x-1
and g(x)=3x+2
.
f(g(x))
, and describe its graph as a transformation of g(x)
. Let f(x)=2x-1
and g(x)=3x+2
.
g(f(x))
, and describe its graph as a transformation of f(x)
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.
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.
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.
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.
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)
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)
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)
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)
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)
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)
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)
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)
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
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)
?
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)
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)
If y=3x-2,x=3t+2
, and t=3k-2
, find an expression for y=f(k)
.
Express y
as a function of k
if y=2x+5,x=\sqrt{3t-1}
, and t=3k-5
.
Review of Combining functions
Composition Analogy
Composition with Set of Points
Composition with Algegra
Decomposition of Functions
ex. Decompose
\displaystyle
y = \frac{1}{\sqrt{1 -x^2}}
ex. Find f, g, h
for f\circ g\circ h(x) = \sqrt{x^2 - 1}
Triple Composition