Download Holt McDougal Algebra 2

Operations
with
Functions
Operations
with
Functions
• How do we add, subtract, multiply,
and divide functions?
• How do we write and evaluate
composite functions?
HoltMcDougal
Algebra2 Algebra 2
Holt
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 McDougal Algebra 2
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 McDougal Algebra 2
Operations with Functions
Reading Math
The composition (f
g of x.”
o
g)(x) or f(g(x)) is read “f of
Caution!
Be careful not to confuse the notation for
multiplication of functions with composition
fg(x) ≠ (f o g)(x)
Holt McDougal Algebra 2
Operations with Functions
Evaluating Composite Functions
x
Given f(x) = 2 and g(x) = 7 – x, find each value.
1. f(g(4))
Step 1 Find g(4)
g(4) = 7 – 4
=3
Step 2 Find f (3)
3
f (3) = 2
=8
So f(g(4)) = 8.
Holt McDougal Algebra 2
g(x) = 7 – x
x
f(x) = 2
Operations with Functions
Evaluating Composite Functions
x
Given f(x) = 2 and g(x) = 7 – x, find each value.
2. g(f(4))
Step 1 Find f (4)
x
4
f(x)
=
2
f(4) = 2
= 16
Step 2 Find g(16)
g(16) = 7 – 16
= –9
So g(f (4)) = –9.
Holt McDougal Algebra 2
g(x) = 7 – x.
Operations with Functions
Evaluating Composite Functions
2
Given f(x) = 2x – 3 and g(x) = x , find each value.
3. f(g(3))
Step 1 Find g(3)
2
g(3) = 3
=9
Step 2 Find f (9)
f(9) = 2(9) – 3
= 15
So f(g(3)) = 15.
Holt McDougal Algebra 2
g(x) = x2
f(x) = 2x – 3
Operations with Functions
Evaluating Composite Functions
2
Given f(x) = 2x – 3 and g(x) = x , find each value.
4. g(f(3))
Step 1 Find f (3)
f(3) = 2(3) – 3
=3
Step 2 Find g(3)
2
g(3) = 3
=9
So g(f (3)) = 9.
Holt McDougal Algebra 2
f(x) = 2x – 3
g(x) = x2
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 McDougal Algebra 2
Operations with Functions
Writing Composite Functions
2
Given f(x) = x – 1 and g(x) = x , write each
1–x
composite function. State the domain of each.
5. f(g(x))
x
f (g(x)) = f (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 McDougal Algebra 2
Operations with Functions
Writing Composite Functions
2
Given f(x) = x – 1 and g(x) = x , write each
1–x
composite function. State the domain of each.
6. g(f (x))
g (f (x)) = g (x2 – 1)
=
Substitute the rule f into g.
(x2 – 1)
1 – (x2 – 1)
x2 – 1
=
2 – x2
Use the rule for g.
Simplify.
The domain of f(g(x)) is x ≠ 1 or {x|x ≠ 1} because
g(1) is undefined.
Holt McDougal Algebra 2
Operations with Functions
Writing Composite Functions
Given f(x) = 3x – 4 and g(x) =
+ 2 , write each
composite. State the domain of each.
7. f(g(x))
+ 2) – 4
Substitute the rule g into f.
=
+6–4
Distribute. Note that x ≥ 0.
=
+2
Simplify.
f(g(x)) = 3(
The domain of f(g(x)) is x ≥ 0 or {x|x ≥ 0}.
Holt McDougal Algebra 2
Operations with Functions
Writing Composite Functions
Given f(x) = 3x – 4 and g(x) =
+ 2 , write each
composite. State the domain of each.
8. g(f (x))
g(f(x)) =
=
Substitute the rule f into g.
Note that x ≥
The domain of g(f(x)) is x ≥
Holt McDougal Algebra 2
4
3
4
3
.
or {x|x ≥
4
3
}.
Operations with Functions
Lesson 14.1 Practice B
Holt McDougal Algebra 2