Download g(x)

6-5
Operations with Functions
You can perform operations on functions in
much the same way that you perform
operations on numbers or expressions. You can
add, subtract, multiply, or divide functions by
operating on their rules.
Holt McDougal Algebra 2
6-5
Operations with Functions
Holt McDougal Algebra 2
6-5
Operations with Functions
Example 1A: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2,
find each function.
(f + g)(x)
(f + g)(x) = f(x) + g(x)
= (4x2 + 3x – 1) + (6x + 2)
2
= 4x + 9x + 1
Holt McDougal Algebra 2
Substitute function rules.
Combine like terms.
6-5
Operations with Functions
Example 1B: Adding and Subtracting Functions
Given f(x) = 4x2 + 3x – 1 and g(x) = 6x + 2,
find each function.
(f – g)(x)
(f – g)(x) = f(x) – g(x)
= (4x2 + 3x – 1) – (6x + 2)
Substitute function rules.
= 4x2 + 3x – 1 – 6x – 2
Distributive Property
= 4x2 – 3x – 3
Combine like terms.
Holt McDougal Algebra 2
6-5
Operations with Functions
Check It Out! Example 1a
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6,
find each function.
(f + g)(x)
(f + g)(x) = f(x) + g(x)
= (5x – 6) + (x2 – 5x + 6)
2
=x
Holt McDougal Algebra 2
Substitute function rules.
Combine like terms.
6-5
Operations with Functions
Check It Out! Example 1b
Given f(x) = 5x – 6 and g(x) = x2 – 5x + 6,
find each function.
(f – g)(x)
(f – g)(x) = f(x) – g(x)
= (5x – 6) – (x2 – 5x + 6)
2
Substitute function rules.
= 5x – 6 – x + 5x – 6
Distributive Property
= –x2 + 10x – 12
Combine like terms.
Holt McDougal Algebra 2
6-5
Operations with Functions
Example 2A: Multiplying and Dividing Functions
Given f(x) = 6x2 – x – 12 and g(x) = 2x – 3,
find each function.
(fg)(x)
(fg)(x) = f(x) ● g(x)
= (6x2 – x – 12) (2x – 3)
Substitute function
rules.
= 6x2 (2x – 3) – x(2x – 3) – 12(2x – 3)
Distributive Property
= 12x3 – 18x2 – 2x2 + 3x – 24x + 36
Multiply.
= 12x3 – 20x2 – 21x + 36
Combine like terms.
Holt McDougal Algebra 2
6-5
Operations with Functions
Example 2B: Multiplying and Dividing Functions
f
g
f
g
( )(x)
( )(x) =
f(x)
g(x)
6x2 – x –12
=
2x – 3
(2x – 3)(3x + 4)
=
2x – 3
(2x – 3)(3x +4)
=
(2x – 3)
= 3x + 4, where x ≠
Holt McDougal Algebra 2
Set up the division as a
rational expression.
Factor completely.
3
Note that x ≠ 2 .
Divide out common
factors.
3
2
Simplify.
6-5
Operations with Functions
Check It Out! Example 2a
Given f(x) = x + 2 and g(x) = x2 – 4, find each
function.
(fg)(x)
(fg)(x) = f(x) ● g(x)
= (x + 2)(x2 – 4)
Substitute function rules.
= x3 + 2x2 – 4x – 8
Multiply.
Holt McDougal Algebra 2
6-5
Operations with Functions
Check It Out! Example 2b
g
(x)
f
g
g(x)
(x)
=
f
f(x)
x2 – 4
=
x+2
(x – 2)(x + 2)
=
x+2
(x – 2)(x + 2)
=
(x + 2)
( )
( )
= x – 2, where x ≠ –2
Holt McDougal Algebra 2
Set up the division as a
rational expression.
Factor completely.
Note that x ≠ –2.
Divide out common
factors.
Simplify.
6-5
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
6-5
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
6-5
Operations with Functions
Reading Math
The composition (f
g of x.”
Holt McDougal Algebra 2
o
g)(x) or f(g(x)) is read “f of
6-5
Operations with Functions
Caution!
Be careful not to confuse the notation for
multiplication of functions with composition
fg(x) ≠ f(g(x))
Holt McDougal Algebra 2
6-5
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 McDougal Algebra 2
f(x) = 2x
6-5
Operations with Functions
Example 3B: Evaluating Composite Functions
Given f(x) = 2x and g(x) = 7 – x, find each
value.
g(f(4))
Step 1 Find f(4)
f(4) = 24
= 16
Step 2
f(x) = 2x
Find g(16)
g(16) = 7 – 16
= –9
So g(f(4)) = –9.
Holt McDougal Algebra 2
g(x) = 7 – x.
6-5
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 McDougal Algebra 2
f(x) = 2x – 3
6-5
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 McDougal Algebra 2
g(x) = x2
6-5
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
6-5
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 McDougal Algebra 2
6-5
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 McDougal Algebra 2
.
}
6-5
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 McDougal Algebra 2
6-5
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 McDougal Algebra 2
4
3
4
3
.
or {x|x ≥
4
3
}.
6-5
Operations with Functions
Lesson Quiz: Part I
Given f(x) = 4x2 – 1 and g(x) = 2x – 1, find
each function or value.
1. (f + g)(x)
4x2 + 2x – 2
2. (fg)(x)
8x3 – 4x2 – 2x + 1
3.
f
g
( )(x)
4. g(f(2))
Holt McDougal Algebra 2
2x + 1
29
6-5
Operations with Functions
Lesson Quiz: Part II
Given f(x) = x2 and g(x) =
, write each
composite function. State the domain of each.
5. f(g(x))
f(g(x)) = x – 1;
{x|x ≥ 1}
6. g(f(x))
{x|x ≤ – 1 or x ≥ 1}
Holt McDougal Algebra 2