Download 2.1

Chapter 2 – Functions and Function Operations
Mini-Lesson
Section 2.1 – Combining Functions
Function notation can be expanded to include notation for the different ways we can combine
functions as described below.
Basic Mathematical Operations on Functions
The basic mathematical operations are: addition, subtraction, multiplication, and division.
When working with function notation, these operations will look like this:
Addition
" # + 2(#)
Subtraction
Multiplication
" # − 2(#)
" # ∙2 #
Division
C D
E D
; g(x) ≠ 0
Many of the problems we will work in this lesson are problems you may already know how to
do. You will just need to get used to some new notation.
We will start with the operations of addition and subtraction.
Problem 1
WORKED EXAMPLE – Adding and Subtracting Functions
Given f (x) = 2x2 + 3x – 5 and g(x) = –x2 + 5x + 1.
a) Find f (x) + g(x)
f ( x) + g ( x) = (2 x 2 + 3 x − 5) + (− x 2 + 5 x + 1)
= 2 x 2 + 3x − 5 − x 2 + 5 x + 1
f ( x) + g ( x) = x 2 + 8 x − 4
b) Find f (x) – g(x)
f ( x) − g ( x) = (2 x 2 + 3x − 5) − (− x 2 + 5 x + 1)
= 2 x 2 + 3x − 5 + x 2 − 5 x − 1
f ( x) − g ( x) = 3 x 2 − 2 x − 6
c) Find f (1) – g(1)
f (1) − g (1) = [2(1) 2 + 3(1) − 5] − [−(1) 2 + 5(1) + 1]
= [2 + 3 − 5] − [−1 + 5 + 1]
= [0] − [5]
f (1) − g (1) = −5
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Chapter 2 – Functions and Function Operations
Problem 2
Mini-Lesson
MEDIA/CLASS EXAMPLE – Adding and Subtracting Functions
Given f (x) = 3x2 + 2x – 1 and g(x)= x2 + 2x + 5:
a) Find f (x) + g(x)
b) Find f (x) – 2g(x)
Problem 3
YOU TRY – Adding and Subtracting Functions
Given f (x) = x2 + 4 and g(x)= x2 + 1, determine each of the following. Show complete work.
a) Find f (2) + g(2)
b) Find 5f (x) – 2g(x)
c) Find 3f (2) – g(2)
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Chapter 2 – Functions and Function Operations
Mini-Lesson
MULTIPLICATION PROPERTY OF EXPONENTS
Let m and n be rational numbers.
To multiply powers of the same base, keep the base and add the exponents:
FG · FI = FGJI
Problem 4
WORKED EXAMPLE – Function Multiplication
a) Given " # = −8# 4 and 2 # = 5# 6 , find " # · 2(#)
f ( x) ⋅ g ( x) = (−8 x 4 )(5 x 3 )
= (−8)(5)( x 4 )( x 3 )
Reorder using Commutative Property
Simplify using the Multiplication
Property of Exponents
= (−40)( x 7 )
f ( x) ⋅ g ( x) = −40 x 7
Final Result
b) Given " # = −3# and 2 # = 4# 1 − # + 8, find " # · 2(#)
f ( x) ⋅ g ( x) = (−3x)(4 x 2 − x + 8)
= (−3x)(4 x 2 ) + (−3x)(− x) + (−3x)(8)
f ( x) ⋅ g ( x) = −12 x 3 + 3x 2 − 24 x
Apply the Distributive Property
Remember the rules of exponents
−3# 4# 1 = −3 4 # 3 # 1
= −12# 6
Final Result
c) Given f (x) = 3x + 2 and g(x) = 2x – 5, find " # · 2 #
f ( x) ⋅ g ( x) = (3x + 2)(2 x − 5)
= (3x)(2 x) + (3x)(−5) + (2)(2 x) + (2)(−5)
= FIRST + OUTER + INNER + LAST
= (6 x 2 ) + (−15 x) + (4 x) + (−10)
f ( x) ⋅ g ( x) = 6 x 2 − 11x − 10
Use the Distributive Property
Remember the rules of exponents
3# 2# ! = ! 3 2 # # = 6# 1
Combine Like Terms
Final Result
Page 60
Chapter 2 – Functions and Function Operations
Problem
5
MEDIA/CLASS EXAMPLE – Function Multiplication
Given " # = 3# + 2 and 2 # = 2# 1 + 3# + 1, find " # · 2(#)
Problem
6
YOU TRY – Function Multiplication
For each of the following, find " # · 2(#)
a) f (x) = 3x – 2 and g(x) = 3x + 2
b) f (x) = 2x2 and g(x) = x3 – 4x + 5
c) f (x) = 4x3 and g(x) = –6x
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Mini-Lesson
Chapter 2 – Functions and Function Operations
Mini-Lesson
DIVISION PROPERTY OF EXPONENTS
Let m, n be rational numbers. To divide powers of the same base, keep the base and subtract
the exponents.
KL
KM
Problem 7
= NOPQ where N ≠ 0
WORKED EXAMPLE – Function Division
For each of the following, find
C D
E D
. Use only positive exponents in your final answer.
a) " # = 15# 3S and 2 # = 3# T
f ( x) 15 x15
=
g ( x) 3x 9
= 5 x15 − 9
= 5x 6
NEGATIVE EXPONENTS
If a ≠ 0 and n is a rational number, then
1
a −n = n
a
b) " # = −4# S and 2 # = 2# U
f ( x) − 4 x 5
=
g ( x)
2 x8
= −2 x 5 − 8
= −2 x − 3
2
=−
x3
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Chapter 2 – Functions and Function Operations
Mini-Lesson
DIVISION OF FUNCTIONS:
When dividing two functions
V D
W D
:
If q(x) is a monomial, we can perform the division of each term of X(#) by Y(#).
If q(x) is not a monomial, we will have to perform the long division algorithm..This is
explained in problem 11 below.
Problem 8
Determine
WORKED EXAMPLE – DIVIDING MONOMIALS
C D
E D
given f(x) = 4x2 + 5x – 3 and g(x) = 2x2. Use only positive exponents in your
final answer.
Since the denominator is a monomial, divide each term in the numerator by the
denominator.
4 x 2 + 5x − 3 4 x 2
5x
3
= 2 + 2 − 2
2
2x
2x
2x
2x
5
3
= 2 +
− 2
2x 2x
Problem 9
MEDIA/CLASS EXAMPLE – DIVIDING MONOMIALS
For each of the following, determine
4
3
2
C D
E D
. Use only positive exponents in your final answer.
a) f(x) = 12x – 8x – 2x + 4x and g(x) = 4x
b) f(x) = 5x3 + 15x2 – 25x and g(x) = 5x3
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Chapter 2 – Functions and Function Operations
Problem 10
YOU TRY – DIVIDING MONOMIALS
For each of the following, determine
3
Mini-Lesson
2
C D
E D
. Use only positive exponents in your final answer.
a) f(x) = 8x – 6x + 4x – 12 and g(x) = 4x
b) f(x) = 9x3 + 6x2 + 3x + 12 and g(x) = 3x2
Problem 11
WORKED EXAMPLE – LONG DIVISION
Given f(x) = x3 – 6x2 – 5 and g(x) = x – 2. Determine
C D
E D
by long division.
STEP 1: Rewrite the terms in the numerator and denominator in descending order, putting
x3 − 6x 2 + 0x − 5
a coefficient of 0 for the missing term:
x−2
STEP 2: Divide the first term of the dividend by the first term of the divisor.
x2
x − 2 x3 − 6x 2 + 0x + 5
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Chapter 2 – Functions and Function Operations
STEP 3: Multiply the divisor by the result just obtained.
x2
x − 2 x3 − 6x 2 + 0x + 5
x3 − 2x 2
STEP 4: Subtract like terms.
x2
x3 − 6x 2 + 0x + 5
x−2
(
− x3 − 2x 2
)
− 4x 2 + 0x
STEP 5: Repeat the previous three steps.
x 2 − 4x
x−2
x3 − 6x 2 + 0x + 5
(
− x3 − 2x 2
)
− 4x 2 + 0x
− 4 x 2 + 8x
STEP 6: Repeat the previous step.
x 2 − 4x − 8
x−2
x3 − 6x 2 + 0x + 5
(
− x3 − 2x 2
)
2
− 4x + 0x
(
− − 4 x 2 + 8x
)
− 8x + 5
− (− 8 x + 16 )
–11 (this is the remainder)
Therefore,
x3 − 6x 2 + 5
11
= x 2 − 4x − 8 −
x−2
x−2
This is the long division algorithm for division of polynomials.
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Mini-Lesson
Chapter 2 – Functions and Function Operations
Problem 12
MEDIA/CLASS EXAMPLE – LONG DIVISION
Divide the following using long division.
4x3 + 2x 2 − 2x − 3
a)
2x + 1
Problem 13
Mini-Lesson
b) (2x3 + 3x2 + 27) ÷!(x + 3)
YOU TRY – LONG DIVISION
Divide the following using long division.
x3 − x 2 − 2x + 6
a)
x−2
b)
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(9x3 – 7x + 2) ÷ (3x + 2)
Chapter 2 – Functions and Function Operations
Mini-Lesson
Functions can be presented in multiple ways including: equations, data sets, graphs, and
applications. If you understand function notation, then the process for working with functions is
the same no matter how the information if presented.
Problem 14
MEDIA/CLASS EXAMPLE – Working with Functions in Table Form
x
f (x)
–3
8
–2
6
0
3
1
2
4
5
5
8
8
11
10
15
12
20
x
g(x)
0
1
2
3
3
5
4
10
5
4
8
2
9
0
11
–2
15
–5
Functions f (x) and g(x) are defined in the tables above. Find the following:
a)
f (1) =
b) g(9) =
c)
f (0) + g(0) =
d) g(5) – f(8) =
e)
" 0 ·2 3 =
Problem 15
YOU TRY – Working with Functions in Table Form
x
f (x)
0
4
1
3
2
–2
3
0
4
1
x
g(x)
0
6
1
–3
2
4
3
–2
4
2
Given the above two tables. Complete the table below. Show your work in the table cell for
each column. The first one is done for you.
x
f (x)+ g(x)
0
f (0) + g(0)
1
2
=4+6
= 10
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3
4
Chapter 2 – Functions and Function Operations
Mini-Lesson
Remember that graphs are just infinite sets of ordered pairs. If you do a little work ahead of
time (as in the example below) then the graphing problems are a lot easier to work with.
Problem 16
YOU TRY – Working with Functions in Graph Form
Use the graph to determine each of the following. Assume integer answers. The graph of g is
the graph in bold.
Complete the following ordered pairs from the graphs above. Use the information to help
you with the problems below. The first ordered pair for each function has been completed
for you.
f: (–7, 2), (–6,
(3,
), (4,
g: (–7, 3), (–6,
(3,
a)
), (4,
), (–5,
), (5,
), (–4,
), (6,
), (–5,
), (5,
), (–3,
), (7,
), (–4,
), (6,
), (–2,
), (–1,
), (0,
), (1,
), (2,
),
), (–2,
), (–1,
), (0,
), (1,
), (2,
),
)
), (–3,
), (7,
)
g(4) = ________________________
b) f (2) =________________
c) g(0) = ________________________
d) f (–6) = ______________
e) If f (x) = 0, x = ________________
f) If g(x) = 0, x = ___________
g) If f (x) = 1, x =____________________
h) If g(x) = – 4, x =__________
i) f (–1) + g(–1) =____________________
j) g(–6) – f (–6) =__________
k) " 1 ∗ 2 −2 =____________________
l)
E \
C P3
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