Honors Math 2
Domains of Composite Functions
Name:
Date:
Remember:
Given functions f and g, the composite function f o g can be described by the following
equation:
( f o g)(x) = f (g(x))
The domain of a composite function f o g includes all of the elements x in the domain of
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g for which g(x) is in the domain of f.
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Example 1:
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Functions
f and g are defined by the tables below.
x
g(x)
x
0
3
0
2
6
3
7
-1
6
10
12
12
15
0
14
f(x)
1
2
8
-4
14
Consider the composite function f o g . What is its domain? Complete the table below to
help you decide.
x
f(g(x))€
1
and g(x) = x − 2 . Find a rule for ( f o g)(x) and give the
x2
domain of the composite function.
Example 2: Let f (x) =
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Classwork/Homework
1.
2
Let:
p(x) = 2x + 1 and q(x) = x − 3 .
And suppose: u(x) = p(q(x)) and v(x) = q(p(x)).
(a) Calculate u(3) and v(3).
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(b) Find the formulas for u(x) and v(x).
(c) Check each formula by evaluating it at x = 3 and compare your answer with
part (a).
(d) Find a formula in terms of x for w(x) = p(p(q(x))).
2.
Composition using function formulas: Suppose f(x) =
1
and g(x) =
x+5
a. Find f(g(4)).
c. Find g(f(4)).
b. Find f(g(x)).
d. Find g(f(x)).
e. Find the domain of f(g(x)).
f. Find the domain of g(f(x)).
x.
3.
Composition using tables: Here are tables for a different pair of functions f(x)
and g(x).
x
–4
–2
0
2
4
f(x)
–2
4
–2
–4
2
x
–4
–2
0
2
4
g(x)
2
0
–2
–4
3
a. Find g(f(4)) or explain why you can’t.
b. Find f(g(4)) or explain why you can’t.
c. Complete this table for g(f(x)).
x
–4
g(f(x))
–2
0
2
4
4.
Composition using graphs: Given f(x) and g(x) graphed below, sketch a graph of
g(f(x)).
f(x)
g(x)
g(f(x))
1
and g(x) = x +1. Find:
x
(a) ( f o g)(x) and its domain.
5. Let f (x) =
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(b) (g o f )(x) and its domain.
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6. For each function, find the natural domain and range.
a.
f ( x) = 3x 2 − 6 x + 9
b.
f ( x) = x + 4 − 2
c.
f ( x) =
2
x+4
7. Are f (x) and g (x) equal functions? Explain your answer.
f (x) =
( x)
2
g ( x) = x
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Some answers
1. a. 𝑢 3 = 13 𝑣 3 = 46 b. 𝑢 𝑥 = 2𝑥 ! − 5 𝑣 𝑥 = 4𝑥 ! + 4𝑥 − 2 d. 4𝑥 ! − 9
2. a.
!
!
!
b. ! c.
!
!!!
d.
!
!!!
e. 𝑥 ≥ 0 f. 𝑥 > −5
3. a. -4 b. undefined c. Outputs should be 0, 3, 0, 2, -4
!
5. a. 𝑓 𝑔 𝑥 = !!! Domain: all real numbers except -1
!
b. 𝑔 𝑓 𝑥 = ! + 1 Domain: all real numbers except 0
6. a. Domain: All real numbers Range: 𝑦 ≥ 6
b. Domain: 𝑥 ≥ −4 Range: 𝑦 ≥ −2
c. Domain: 𝑥 ≠ −4 Range: 𝑦 ≠ 0
7. No, the functions have different domain. 𝑓 𝑥 domain: 𝑥 ≥ 0