key concept 1. example 1 operations with functions a. given f (x) = x 2 – 2x, g (x) = 3x – 4,...
TRANSCRIPT
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 oftwo functions
= (x 2 – 2x) + (3x – 4)
f (x) = x 2 – 2x;
g (x) = 3x – 4
= x 2 + 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
Simplify. The domain of f and h are both so the domain of (f – h) is
Answer:
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 + 8xSimplify.
The domain of f and g are both so the domain of (f ● g) is
Answer:
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 domains of h and f are both (–∞, ∞), but x = 0 or
x = 2 yields a zero in the denominator of . So, the
domain of (–∞, 0) (0, 2) (2, ∞).
Answer: 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).
Replace g (x) with x + 3= f (x + 3)
Substitute x + 3 for x in f (x).
= 2(x + 3)2 – 1
Answer: [f ○ g](x) = 2x 2 + 12x + 17
Expand (x +3)2= 2(x 2 + 6x + 9) – 1
Simplify.= 2x 2 + 12x + 17
Compose Two Functions
B. Given f (x) = 2x2 – 1 and g (x) = x + 3, find [g ○ f](x).
Substitute 2x 2 – 1 for
x in g (x).= (2x
2 – 1) + 3
Simplify= 2x 2 + 2
Answer: [g ○ f](x) = 2x 2 + 2
Compose Two Functions
Evaluate the expression you wrote in part A for x = 2.
Answer: [f ○ g](2) = 49
C. Given f (x) = 2x 2 – 1 and g (x) = x + 3, find [f ○ g](2).
[f ○ g](2) = 2(2)2 + 12(2) + 17 Substitute 2 for x.
= 49 Simplify.
A. 2x 2 + 11; 4x
2 – 12x + 13; 23
B. 2x 2 + 11; 4x
2 – 12x + 5; 23
C. 2x 2 + 5; 4x
2 – 12x + 5; 23
D. 2x 2 + 5; 4x
2 – 12x + 13; 23
Find for f (x) = 2x – 3 and g (x) = 4 + x
2.
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).
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
Replace g (x) with (x – 1)2.
Substitute (x – 1)2 for x in
f (x).
Simplify.
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:
h
Decompose a Composite Function
h (x) = 3x2 – 12x + 12 Notice that h is factorable.
= 3(x2 – 4x + 4) or 3(x – 2)
2 Factor.
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.
One way to write h (x) as a composition is to let f (x) = 3x2 and g (x) = x – 2.
Sample answer: g (x) = x – 2 and f (x) = 3x 2
Decompose a Composite Function
A.
B.
C.
D.