Five-Minute Check
Then/Now
New Vocabulary
Key Concept: Operations with Functions
Example 1: Operations with Functions
Key Concept: Composition of Functions
Example 2: Compose Two Functions
Example 3: Find a Composite Function with a Restricted
Domain
Example 4: Decompose a Composite Function
Example 5: Real-World Example: Compose Real-World
Functions
Use the graph of y = x 2 to describe the graph of the
related function y = 0.5x 2.
A. The parent graph is translated up 0.5 units.
B. The parent graph is expanded horizontally by
a factor of 0.5.
C. The parent graph is compressed vertically.
D. The parent graph is translated down 0.5 units.
Use the graph of y = x 2 to describe the graph of the
related function y = (x – 4)2 – 3.
A. The parent graph is translated left 3 units and
up 4 units.
B. The parent graph is translated right 3 units
and down 4 units.
C. The parent graph is translated left 4 units and
down 3 units.
D. The parent graph is translated right 4 units
and down 3 units.
A.
C.
B.
D.
Identify the parent function f(x) if
and describe how the graphs of g(x) and f(x) are
related.
A. f(x) = x; f(x) is translated left 4 units.
B. f(x) = |x|; f(x) is translated right 4 units.
C.
D.
You evaluated functions. (Lesson 1-1)
• Perform operations with functions.
• Find compositions of functions.
• composition
Operations with Functions
A. Given f(x) = x 2 – 2x, g(x) = 3x – 4, and
h(x) = –2x 2 + 1, find the function and domain for
(f + g)(x).
(f + g)(x) = f(x) + g(x)
Definition of sum of
two functions
= (x 2 – 2x) + (3x – 4)
f(x) = x 2 – 2x;
g(x) = 3x – 4
= x2 + x – 4
Simplify.
The domain of f and g are both
so the domain
of (f + g) is
Answer:
Operations with Functions
B. Given f(x) = x 2 – 2x, g(x) = 3x – 4, and
h(x) = –2x 2 + 1, find the function and domain for
(f – h)(x).
(f – h)(x) = f(x) – h(x)
Definition of difference
of two functions
= (x 2 – 2x) – (–2x 2 + 1) f(x) = x 2 – 2x;
h(x) = –2x 2 + 1
= 3x 2 – 2x – 1
The domain of f and h are both
of (f – h) is
Answer:
Simplify.
so the domain
Operations with Functions
C. Given f(x) = x 2 – 2x, g(x) = 3x – 4, and
h(x) = –2x 2 + 1, find the function and domain for
(f ● g)(x).
(f ● g)(x) = f(x) g(x)
Definition of product of
two functions
= (x 2 – 2x)(3x – 4)
f(x) = x 2 – 2x;
g(x) = 3x – 4
= 3x 3 – 10x 2 + 8x
Simplify.
The domain of f and g are both
of (f ● g) is
Answer:
so the domain
Operations with Functions
D. Given f(x) = x 2 – 2x, g(x) = 3x – 4, and
h(x) = –2x 2 + 1, find the function and domain for
Definition of quotient of two
functions
f(x) = x 2 – 2x; h(x) = –2x 2 + 1
Operations with Functions
The domain of h and f are both, but x = 0 or x = 2
yields a zero in the denominator of
domain of
Answer:
. So, the
(–∞, 0) (0, 2)  (2, ∞).
D = (–∞, 0) (0, 2)  (2, ∞)
Find (f + g)(x), (f – g)(x), (f ● g)(x), and
for
f(x) = x 2 + x, g(x) = x – 3. State the domain of each
new function.
A.
B.
C.
D.
Compose Two Functions
A. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [f ○ g](x).
= f(x + 3)
Replace g(x) with x + 3
= 2(x + 3)2 – 1
Substitute x + 3 for
x in f(x).
= 2(x 2 + 6x + 9) – 1 Expand (x +3)2
= 2x 2 + 12x + 17
Simplify.
Answer: [f ○ g](x) = 2x 2 + 12x + 17
Compose Two Functions
B. Given f(x) = 2x2 – 1 and g(x) = x + 3, find [g ○ f](x).
Answer:
= (2x 2 – 1) + 3
Substitute 2x 2 – 1 for
x in g(x).
= 2x 2 + 2
Simplify
[g ○ f](x) = 2x 2 + 2
Compose Two Functions
C. Given f(x) = 2x 2 – 1 and g(x) = x + 3, find [f ○ g](2).
Evaluate the expression you wrote in part A for x = 2.
[f ○ g](2) = 2(2)2 + 12(2) + 17
= 29
Answer: [f ○ g](2) = 29
Substitute 2 for x.
Simplify.
Find
and g(x) = 4 + x 2.
A. 2x 2 + 5; 4x 2 – 12x + 13; 23
B. 2x 2 + 11; 4x 2 – 12x + 13; 23
C. 2x 2 + 5; 4x 2 – 12x + 5; 23
D. 2x 2 + 5; 4x 2 – 12x + 13; 13
for f(x) = 2x – 3
Find a Composite Function with a Restricted
Domain
A. Find
.
Find a Composite Function with a Restricted
Domain
To find
, you must first be able to find g(x) = (x – 1) 2,
which can be done for all real numbers. Then you must
be able to evaluate
for each of these
g(x)-values, which can only be done when g(x) > 1.
Excluding from the domain those values for which
0 < (x – 1) 2 <1, namely when 0 < x < 1, the domain of
f ○ g is (–∞, 0]  [2, ∞). Now find [f ○ g](x).
Find a Composite Function with a Restricted
Domain
Replace g(x) with (x – 1)2.
Substitute (x – 1)2 for x in
f(x).
Simplify.
Notice that
is not defined for 0 < x < 2.
Because the implied domain is the same as the
domain determined by considering the domains of
f and g, we can write the composition as
for (–∞, 0] [2, ∞).
Find a Composite Function with a Restricted
Domain
Answer:
for (–∞, 0] [2, ∞).
Find a Composite Function with a Restricted
Domain
B. Find f ○ g.
Find a Composite Function with a Restricted
Domain
To find f ○ g, you must first be able to find
,
which can be done for all real numbers x such that x2  1.
Then you must be able to evaluate
for each of
these g(x)-values, which can only be done when g(x)  0.
Excluding from the domain those values for which
0  x 2 < 1, namely when –1 < x < 1, the domain of f ○ g is
(–∞, –1) (1, ∞). Now find [f ○ g](x).
Find a Composite Function with a Restricted
Domain
Find a Composite Function with a Restricted
Domain
Answer:
Find a Composite Function with a Restricted
Domain
Check Use a graphing calculator to check this result.
Enter the function as
. The graph appears
to have asymptotes at x = –1 and x = 1. Use the
TRACE feature to help determine that the domain of
the composite function does not include any values in
the interval [–1, 1].
Find a Composite Function with a Restricted
Domain
Find f ○ g.
A. D = (–∞, –1)  (–1, 1)  (1, ∞);
B. D = [–1, 1];
C. D = (–∞, –1)  (–1, 1)  (1, ∞);
D. D = (0, 1);
Decompose a Composite Function
A. Find two functions f and g such that
when
. Neither function may be the
identity function f(x) = x.
Decompose a Composite Function
Sample answer:
Decompose a Composite Function
B. Find two functions f and g such that
when h(x) = 3x 2 – 12x + 12. Neither function may
be the identity function f(x) = x.
h(x) = 3x2 – 12x + 12
Notice that h is
factorable.
= 3(x2 – 4x + 4) or 3(x – 2) 2 Factor.
One way to write h(x) as a composition is to let
f(x) = 3x2 and g(x) = x – 2.
Decompose a Composite Function
Sample answer: g(x) = x – 2 and f(x) = 3x 2
A.
B.
C.
D.
Compose Real-World Functions
A. COMPUTER ANIMATION An animator starts with
an image of a circle with a radius of 25 pixels. The
animator then increases the radius by 10 pixels per
second. Find functions to model the data.
The length r of the radius increases at a rate of 10 pixels
per second, so R(t) = 25 + 10t, where t is the time in
seconds and t  0. The area of the circle is  times the
square of the radius. So, the area of the circle is
A(R) = R 2.
So, the functions are R(t) = 25 + 10t and A(R) = R 2.
Answer: R(t) = 25 + 10t; A(R) = R 2
Compose Real-World Functions
B. COMPUTER ANIMATION An animator starts with
an image of a circle with a radius of 25 pixels. The
animator then increases the radius by 10 pixels per
second. Find A ○ R. What does the function
represent?
A ○ R = A[R(t)]
Definition of A ○ R
=A(25 + 10t)
Replace R(t) with
25 + 10t.
= (25 + 10t)2
Substitute (25 + 10t)
for R in A(R).
= 100t 2 + 500t + 625 Simplify.
Compose Real-World Functions
So, A ○ R = 100t 2 + 500t + 625. The composite
function models the area of the circle as a function of
time.
Answer: A ○ R = 100t 2 + 500t + 625 ; the
function models the area of the circle as a
function of time.
Compose Real-World Functions
C. COMPUTER ANIMATION An animator starts with
an image of a circle with a radius of 25 pixels. The
animator then increases the radius by 10 pixels per
second. How long does it take for the circle to
quadruple its original size?
The initial area of the circle is  ● 25 2 = 625 pixels. The
circle will be four times its original size when
[A ◦ R ](t) = 100t 2 + 500t + 625 = 4(625) = 2500.
Solve for t to find that t = 2.5 or –7.5 seconds. Because a
negative t-value is not part of the domain of R(t), it is
also not part of the domain of the composite function.
The area will quadruple after 2.5 seconds.
Answer: 2.5 seconds
BUSINESS A satellite television company offers a 20% discount on
the installation of any satellite television system. The company also
advertises $50 in coupons for the cost of equipment. Find [c ◦ d](x)
and [d ◦ c](x). Which composition of the coupon and discount results
in the lower price? Explain.
A. [c ◦ d](x) = 0.80x – 40; [d ◦ c](x) = 0.80x – 50; Sample answer:
[d ◦ c](x) represents the cost of installation using the coupon and
then the discount results in the lower cost.
B.
[c ◦ d](x) = 0.80x – 40; [d ◦ c](x) = 0.80x – 50; Sample answer:
[c ◦ d](x) represents the cost of installation using the discount and
then the coupon results in the lower cost.
C.
[c ◦ d](x) = 0.80x – 50; [d ◦ c](x) = 0.80x – 40; Sample answer:
[c ◦ d](x) represents the cost of installation using the discount and
then the coupon results in the lower cost.
D.
[c ◦ d](x) = 0.80x – 50; [d ◦ c](x) = 0.80x – 40; Sample answer:
[c ◦ d](x) represents the cost of installation using the coupon and
then the discount results in the lower cost.