5.1 Combining Functions Perform arithmetic operations on functions Review function notation Perform composition of functions

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

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.

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

Example: Evaluating combinations of functions (2 of 3) Solution f(4) = 9, g(4) = 2 (f + g)(4) = 9 + 2 = 11 f(−2) = −3, g(−2) is undefined, so (f + g) is undefined

Example: Evaluating combinations of functions (3 of 3) f(1) = 3, g(1) = 1 (fg)(3) = 3(1) = 3

Example: Performing arithmetic operations on functions symbolically (1 of 6)

Example: Performing arithmetic operations on functions symbolically (2 of 6)

Example: Performing arithmetic operations on functions symbolically (3 of 6)

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).

Example: Performing arithmetic operations on functions symbolically (5 of 6)

Example: Performing arithmetic operations on functions symbolically (6 of 6)

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 − 12 + 2 = 2 b. g(k) = 3k² − 6k + 2

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 − 12 + 2 = 3x² + 6x + 2

Composition of Functions (1 of 2)

Domain of Composition of Functions

Composition of Functions (2 of 2)

Example: Evaluating a composite function symbolically (1 of 5)

Example: Evaluating a composite function symbolically (2 of 5)

Example: Evaluating a composite function symbolically (3 of 5)

Example: Evaluating a composite function symbolically (4 of 5)

Example: Evaluating a composite function symbolically (5 of 5)