9-4
with
Functions
9-4 Operations
Operations
with
Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra2
Holt
Algebra
2
9-4
Operations with Functions
Warm Up
Simplify. Assume that all expressions
are defined.
1. (4x + 5) – (x2 + 3x – 4)
–x2 + x + 9
3.
x–3
x–2
Holt Algebra 2
9-4
Operations with Functions
Objectives
Write and evaluate composite
functions.
Alg 2 Standard: 24.0
Students solve problems involving functional
concepts, such as composition, defining the
inverse function and performing arithmetic
operations on functions.
Holt Algebra 2
9-4
Operations with Functions
Vocabulary
composition of functions
Holt Algebra 2
9-4
Operations with Functions
Another function operation uses the output from
one function as the input for a second function.
This operation is called the composition of
functions.
Holt Algebra 2
9-4
Operations with Functions
The order of function operations is the same as the
order of operations for numbers and expressions. To
find f(g(3)), evaluate g(3) first and then substitute the
result into f.
Holt Algebra 2
9-4
Operations with Functions
Caution!
Be careful not to confuse the notation for
multiplication of functions with composition
fg(x) ≠ f(g(x))
Holt Algebra 2
9-4
Operations with Functions
Example 3A: Evaluating Composite Functions
Given f(x) = 2x and g(x) = 7 – x, find each
value.
f(g(4))
Step 1 Find g(4)
g(4) = 7 – 4
g(x) = 7 – x
=3
Step 2 Find f(3)
3
f(3) = 2
=8
So f(g(4)) = 8.
Holt Algebra 2
f(x) = 2x
9-4
Operations with Functions
Check It Out! Example 3a
Given f(x) = 2x – 3 and g(x) = x2, find each
value.
f(g(3))
Step 1 Find g(3)
g(3) = 32
g(x) = x2
=9
Step 2 Find f(9)
f(9) = 2(9) – 3
= 15
So f(g(3)) = 15.
Holt Algebra 2
f(x) = 2x – 3
9-4
Operations with Functions
Check It Out! Example 3b
Given f(x) = 2x – 3 and g(x) = x2, find each
value.
g(f(3))
Step 1 Find f(3)
f(3) = 2(3) – 3
f(x) = 2x – 3
=3
Step 2 Find g(3)
2
g(3) = 3
=9
So g(f(3)) = 9.
Holt Algebra 2
g(x) = x2
9-4
Operations with Functions
You can use algebraic expressions as well as
numbers as inputs into functions. To find a rule
for f(g(x)), substitute the rule for g into f.
Holt Algebra 2
9-4
Operations with Functions
Example 4A: Writing Composite Functions
x
2
Given f(x) = x – 1 and g(x) =
, write
1–x
each composite function. State the domain of
each.
f(g(x))
f(g(x)) = f(
=(
=
x
1–x
x
1–x
)
)2 – 1
–1 + 2x
(1 – x)2
Substitute the rule g into f.
Use the rule for f. Note that
x ≠ 1.
Simplify.
The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because
g(1) is undefined.
Holt Algebra 2
9-4
Operations with Functions
Example 4B: Writing Composite Functions
x
2
Given f(x) = x – 1 and g(x) =
, write
1–x
each composite function. State the domain of
each.
g(f(x))
2
g(f(x)) = g(x – 1)
=
(x2 – 1)
2
1 – (x – 1)
x2 – 1
=
2 – x2
Substitute the rule f into g.
Use the rule for g.
Simplify. Note that x ≠
The domain of g(f(x)) is x ≠
or {x|x ≠
because f(
) = 1 and g(1) is undefined.
Holt Algebra 2
.
}
9-4
Operations with Functions
Check It Out! Example 4a
Given f(x) = 3x – 4 and g(x) =
+ 2 , write
each composite. State the domain of each.
f(g(x))
f(g(x)) = 3(
+ 2) – 4
Substitute the rule g into f.
=
+6–4
Distribute. Note that x ≥ 0.
=
+2
Simplify.
The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.
Holt Algebra 2
9-4
Operations with Functions
Check It Out! Example 4b
Given f(x) = 3x – 4 and g(x) =
+ 2 , write
each composite. State the domain of each.
g(f(x))
g(f(x)) =
=
Substitute the rule f into g.
Note that x ≥
The domain of g(f(x)) is x ≥
Holt Algebra 2
4
3
4
3
.
or {x|x ≥
4
3
}.
9-4
Operations with Functions
Lesson Quiz
Given f(x) = x and g(x) =
, write each
composite function. State the domain of each.
f(g(x)) = x – 1;
5. f(g(x))
2
{x|x ≥ 1}
6. g(f(x))
{x|x ≤ – 1 or x ≥ 1}
Given f(x) = 4x2 – 1 and g(x) = 2x – 1, find
each function or value.
4. g(f(2))
Holt Algebra 2
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