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5.1 Combining Functions Perform arithmetic operations on functions
Review function notation Perform composition of functions
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Operations on Functions
If f(x) and g(x) both exist, the sum, difference, product, quotient and composition of two functions f and g are defined by
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Operations on Functions and Domains
The domains of the sum, difference, and product of f and g include x-values that are in both the domain of f and the domain of g.
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Example: Evaluating combinations of functions (1 of 3)
If possible, use the graph representation of f and g to evaluate (f + g)(4), (f − g)(−2), (fg)(1), and
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Example: Evaluating combinations of functions (2 of 3)
Solution f(4) = 9, g(4) = 2 (f + g)(4) = = 11 f(−2) = −3, g(−2) is undefined, so (f + g) is undefined
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Example: Evaluating combinations of functions (3 of 3)
f(1) = 3, g(1) = 1 (fg)(3) = 3(1) = 3
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Example: Performing arithmetic operations on functions symbolically (1 of 6)
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Example: Performing arithmetic operations on functions symbolically (2 of 6)
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Example: Performing arithmetic operations on functions symbolically (3 of 6)
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Example: Performing arithmetic operations on functions symbolically (4 of 6)
b. (fg)(0) is not defined, since 0 is not in the domain of f(x).
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Example: Performing arithmetic operations on functions symbolically (5 of 6)
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Example: Performing arithmetic operations on functions symbolically (6 of 6)
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Example: Evaluating function notation (1 of 2)
Let g(x) = 3x² − 6x + 2. Evaluate each expression. a. g(2) b. g(k) c. g(x²) d. g(x + 2) Solution a. g(2) = 3(2)² − 6(2) + 2 = 12 − = 2 b. g(k) = 3k² − 6k + 2
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Example: Evaluating function notation (2 of 2)
d. g(x + 2) = 3(x + 2)² − 6(x + 2) + 2 = 3(x² + 4x + 4) − 6(x + 2) + 2 = 3x² + 12x + 12 − 6x − = 3x² + 6x + 2
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Composition of Functions (1 of 2)
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Domain of Composition of Functions
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Composition of Functions (2 of 2)
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Example: Evaluating a composite function symbolically (1 of 5)
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Example: Evaluating a composite function symbolically (2 of 5)
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Example: Evaluating a composite function symbolically (3 of 5)
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Example: Evaluating a composite function symbolically (4 of 5)
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Example: Evaluating a composite function symbolically (5 of 5)
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