2. CHAPTER 2: More on Functions
2.1 Increasing, Decreasing, and Piecewise
Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
2.4 Symmetry and Transformations
2.5 Variation and Applications
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3. 2.2 The Algebra of Functions
Find the sum, the difference, the product, and the quotient of two functions, and determine the domains of the resulting functions.
Find the difference quotient for a function.
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4. Sums, Differences, Products, and Quotients of Functions
If f and g are functions and x is in the domain of each function, then
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Example
Given that f(x) = x + 2 and g(x) = 2x + 5, find each of the following.
a) (f + g)(x) b) (f + g)(5)
Solution: a)
b) We can find (f + g)(5) provided 5 is in the domain of each function. This is true.
f(5) = = 7 g(5) = 2(5) + 5 = 15
(f + g)(5) = f(5) + g(5) = = 22 or
(f + g)(5) = 3(5) + 7 = 22
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Another Example
Given that f(x) = x2 + 2 and g(x) = x  3, find each of the following.
a) The domain of f + g, f  g, fg, and f/g
b) (f  g)(x)
c) (f/g)(x)
Solution:
a) The domain of f is the set of all real numbers. The domain of g is also the set of all real numbers. The domains of f + g, f  g, and fg are the set of numbers in the intersection of the domains—that is, the set of numbers in both domains, or all real numbers.
For f/g, we must exclude 3, since g(3) = 0.
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7. Another Example continued
b) (f  g)(x) = f(x)  g(x) = (x2 + 2)  (x  3)
= x2  x + 5
c) (f/g)(x) =
Remember to add the stipulation that x  3,
since 3 is not in the domain of (f/g)(x).
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Difference Quotient
The ratio below is called the difference quotient, or average rate of change.
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Example
For the function f given by f (x) = 5x  1, find the
difference quotient
Solution: We first find f (x + h):
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Example continued
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Another Example
For the function f given by f (x) = x2 + 2x  3, find
the difference quotient.
Solution: We first find f (x + h):
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Example continued
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