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Translations and
Combinations
Section P.7
Part 5
Arithmetic Combinations of
Functions:
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When you have two real numbers, you can
combine them to form other real numbers by the
operations of:
Addition
Subtraction
Multiplication
Division
Two functions can also be combined using these
operations.
2
Sum, Difference, Product, and
Quotient of Functions
Let f and g be two functions with
overlapping domains.
Then, for all x common to both
domains, the sum, difference,
product, and quotient of f and g are
defined as follows…
3
(1)
(2)
(3)
(4)
Sum:
Difference:
Product:
Quotient:
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) – g(x)
(fg)(x) = f(x)·g(x)
(f/g)(x) = f(x)/g(x)
4
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Example: Finding Arithmetic
Combinations
For the following
functions f and g,
find the sum,
difference,
product, and
quotient of f and g.
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Also find the domain
of each.
f x   3 x  1
gx   2 x  4 x  5
2
5
Example:
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f x   3 x  1
gx   2 x  4 x  5
2
To find the sum of two functions, you
need to add or subtract like terms.
(f + g)(x) = f(x) + g(x)
= (-3x-1) + (2x²-4x+5)
= 2x² - 7x + 4
Domain: (-∞,∞)
6
Example:
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f x   3 x  1
gx   2 x  4 x  5
2
To find the difference of two functions,
you need to add or subtract like terms.
(f - g)(x) = f(x) - g(x)
= (-3x-1) - (2x²-4x+5)
= -2x² + x - 6
Domain: (-∞,∞)
7
Example:
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f x   3 x  1
gx   2 x  4 x  5
2
To find the product of two functions, you
can use the distributive property, if
possible.
(fg)(x) = f(x)·g(x)
= (-3x-1)(2x²-4x+5)
= -6x³ + 10x² - 11x - 5
Domain: (-∞,∞)
8
Example:
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f x   3 x  1
gx   2 x  4 x  5
2
To find the quotient of two functions, you
need to divide the two and reduce, if
possible.
(f/g)(x) = f(x) ÷ g(x)
= -3x-1_
2x²-4x+5
This cannot be reduced
Domain: (-∞,∞)
9
However, sometimes the domain
does not consist of all real numbers.
Any restrictions on the domains of f
and g must be considered when
performing these operations.
10
Example: Finding Arithmetic
Combinations
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For the following functions f and g, find the sum,
difference, product, and quotient of f and g.
Also find the domain of each.
3
f x   5 x  x
4
2
gx   5  2 x
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Remember: you cannot have a square root of a
negative number.
The domain will be limited.
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Example:
3
f x   5 x  x
4
2
gx  
5  2x
(f + g)(x) = f(x) + g(x)
= 5x² - ¾x + √5-2x
Domain: (-∞,5/2]
12
Example:
3
f x   5 x  x
4
2
gx  
5  2x
(f - g)(x) = f(x) - g(x)
= 5x² - ¾x - √5-2x
Domain: (-∞,5/2]
13
Example:
3
f x   5 x  x
4
2
gx  
5  2x
(fg)(x) = f(x) · g(x)
= (5x² - ¾x)√5-2x
Domain: (-∞,5/2]
14
Example:
3
f x   5 x  x
4
2
gx  
5  2x
(f/g)(x) = f(x) ÷ g(x)
= (5x² - ¾x)
√5 - 2x
Domain: (-∞,5/2)
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Homework:
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Finish Worksheet from class.
Pg 93 – 95
15 – 42 (mult. of 3), 63, 64
16