• composition of functions
Add and Subtract Functions
A. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find
(f + g)(x).
(f + g)(x) = f(x) + g(x)
Addition of
functions
= (3x2 + 7x) + (2x2 – x – 1)
f(x) = 3x2 + 7x
and
g(x) = 2x2 – x – 1
= 5x2 + 6x – 1
Simplify.
Answer: 5x2 + 6x – 1
A. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4,
find (f + g)(x).
A. 5x2 + 8x – 2
B. 5x2 + 8x + 6
C. x2 – 2x – 6
D. 5x4 + 8x2 – 2
A.
B.
C.
D.
A
B
C
D
Add and Subtract Functions
B. Given f(x) = 3x2 + 7x and g(x) = 2x2 – x – 1, find
(f – g)(x).
(f – g)(x) = f(x) – g(x)
Subtraction of
functions
= (3x2 + 7x) – (2x2 – x – 1)
f(x) = 3x2 + 7x
and
g(x) = 2x2 – x – 1
= x2 + 8x + 1
Simplify.
Answer: x2 + 8x + 1
B. Given f(x) = 2x2 + 5x + 2 and g(x) = 3x2 + 3x – 4,
find (f – g)(x).
A. –x2 + 2x + 5
A
0%
B
D. –x + 2x + 6
0%
2
A
B
C
0%
D
D
C. –x2 + 2x – 2
A.
B.
C.
0%
D.
C
B. x2 – 2x – 6
Multiply and Divide Functions
A. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find
(f ● g)(x).
(f ● g)(x) = f(x) ● g(x)
Product of
functions
= (3x2 – 2x + 1)(x – 4)
Substitute.
= 3x2(x – 4) – 2x(x – 4) + 1(x – 4) Distributiv
Property
= 3x3 – 12x2 – 2x2 + 8x + x – 4
Distributive
= 3x3 – 14x2 + 9x – 4
Simplify.
Property
Answer: 3x3 – 14x2 + 9x – 4
A. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find
(f ● g)(x).
A. 2x3 + 3x2 – x + 2
D. 2x + 7x + 7x + 2
0%
B
0%
2
A
3
A
B
C
0%
D
D
C. 2x3 + 7x2 + 5x – 2
A.
B.
C.
0%
D.
C
B. 2x3 + 3x – 2
Multiply and Divide Functions
B. Given f(x) = 3x2 – 2x + 1 and g(x) = x – 4, find
Division of functions
f(x) = 3x2 – 2x + 1 and
g(x) = x – 4
Answer:
Multiply and Divide Functions
Since 4 makes the denominator 0, it is excluded from
the domain of
B. Given f(x) = 2x2 + 3x – 1 and g(x) = x + 2, find
.
A.
0%
B
D.
A
0%
A
B
C
0%
D
D
C.
A.
B.
C.
0%
D.
C
B.
Compose Functions
A. If f(x) = (2, 6), (9, 4), (7, 7), (0, –1) and
g(x) = (7, 0), (–1, 7), (4, 9), (8, 2), find [f ○ g](x) and
[g ○ f](x).
To find f ○ g, evaluate g(x) first. Then use the range of g
as the domain of f and evaluate f(x).
f[g(7)] = f(0) or –1
g(7) = 0
f[g(–1)] = f(7) or 7
g(–1) = 7
f[g(4)] = f(9) or 4
g(4) = 9
f[g(8)] = f(2) or 6
g(8) = 2
Answer: f ○ g = {(7, –1), (–1, 7), (4, 4), (8, 6)}
Compose Functions
To find g ○ f, evaluate f(x) first. Then use the range of f
as the domain of g and evaluate g(x).
g[f(2)] = g(6)
g(6) is undefined.
g[f(9)] = g(4) or 9
f(9) = 4
g[f(7)] = g(7) or 0
f(7) = 7
g[f(0)] = g(–1) or 7
f(0) = –1
Answer: Since 6 is not in the domain of g, g ○ f is
undefined for x = 2.
g ○ f = {(9, 9), (7, 0), (0, 7)}
A. If f(x) = {(1, 2), (0, –3), (6, 5), (2, 1)} and g(x) = {(2, 0),
(–3, 6), (1, 0), (6, 7)}, find f ○ g and g ○ f.
A. f ○ g = {(2, –3), (–3, 5), (1, –3)};
g ○ f = {(1, 0), (0, 6), (2, 0)}
B. f ○ g = {(1, 0), (0, 6), (2, 0)};
g ○ f = {(2, –3), (–3, 5), (1, –3)}
C. f ○ g = {(–3, 2), (5, –3), (–3, 1)};
g ○ f = {(0, 1), (6, 0), (0, 2)}
D. f ○ g = {(0, 1), (6, 0), (0, 2)};
g ○ f = {(–3, 2), (5, –3), (–3, 1)}
A.
B.
C.
D.
A
B
C
D
Compose Functions
B. Find [f ○ g](x) and [g ○ f](x) for f(x) = 3x2 – x + 4
and g(x) = 2x – 1.
[f ○ g](x) = f[g(x)]
Composition of
functions
= f(2x – 1)
Replace g(x)
with 2x – 1.
= 3(2x – 1)2 – (2x – 1) + 4
Substitute 2x – 1
for x in f(x).
Compose Functions
= 3(4x2 – 4x + 1) – 2x + 1 + 4
Evaluate
(2x – 1)2.
= 12x2 – 14x + 8
Simplify.
[g ○ f](x) = g[f(x)]
= g(3x2 – x + 4)
Composition
of functions
Replace
f(x) with
3x2 – x + 4.
Compose Functions
= 2(3x2 – x + 4) – 1
Substitute
3x2 – x + 4
for x in g(x).
= 6x2 – 2x + 7
Simplify.
Answer: So, [f ○ g](x) = 12x2 – 14x + 8 and
[g ○ f](x) = 6x2 – 2x + 7.
B. Find [f ○ g](x) and [g ○ f](x) for f(x) = x2 + 2x + 3
and g(x) = x + 5.
A. [f ○ g](x) = x2 + 2x + 8
[g ○ f](x) = x2 + 12x + 38
B. [f ○ g](x) = x2 + 12x + 38
[g ○ f](x) = x2 + 12x + 38
C. [f ○ g](x) = x2 + 12x + 38
[g ○ f](x) = x2 + 2x + 8
D. [f ○ g](x) = x3 + 7x2 + 13x + 15
[g ○ f](x) = x3 + 7x2 + 13x + 15
A.
B.
C.
D.
A
B
C
D