5 1 Combining Functions Perform arithmetic operations on

5. 1 Combining Functions • Perform arithmetic operations on functions • Review function notation • Perform composition of functions Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

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 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

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. Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

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 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

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 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating combinations of functions (3 of 3) f(1) = 3, g(1) = 1 (fg)(3) = 3(1) = 3 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Performing arithmetic operations on functions symbolically (1 of 6) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Performing arithmetic operations on functions symbolically (2 of 6) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Performing arithmetic operations on functions symbolically (3 of 6) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

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). Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Performing arithmetic operations on functions symbolically (5 of 6) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Performing arithmetic operations on functions symbolically (6 of 6) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating function notation (1 of 2) Let g(x) = 3 x² − 6 x + 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) = 3 k² − 6 k + 2 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating function notation (2 of 2) d. g(x + 2) = 3(x + 2)² − 6(x + 2) + 2 = 3(x² + 4 x + 4) − 6(x + 2) + 2 = 3 x² + 12 x + 12 − 6 x − 12 + 2 = 3 x² + 6 x + 2 Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Composition of Functions (1 of 2) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Domain of Composition of Functions Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Composition of Functions (2 of 2) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating a composite function symbolically (1 of 5) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating a composite function symbolically (2 of 5) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating a composite function symbolically (3 of 5) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating a composite function symbolically (4 of 5) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

Example: Evaluating a composite function symbolically (5 of 5) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved