5 1 Combining Functions Perform arithmetic operations on
5. 1 Combining Functions ♦ Perform arithmetic operations on functions ♦ Perform composition of functions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Five Ways of Combining Two Functions f and g • Addition • • Subtraction • • Multiplication • • Division • • Composition • Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 2
Definition-Addition If f(x) and g(x) both exist, the sum, of two functions f and g are defined by Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 3
Example of Addition of Functions: Let f(x) = x 2 + 2 x and g(x) = 3 x - 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 4
Definitions-Subtraction If f(x) and g(x) both exist, the difference of two functions f and g are defined by Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 5
Example of Subtraction of Functions: Let f(x) = x 2 + 2 x and g(x) = 3 x 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 6
Examples of Evaluating Combinations of Functions – Using Symbolic Representations Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 7
Definitions-Multiplication If f(x) and g(x) both exist, the product of two functions f and g are defined by Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 8
Example of Multiplication of Functions: Let f(x) = x 2 + 2 x and g(x) = 3 x 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 9
Definitions-Division If f(x) and g(x) both exist, quotient of two functions f and g are defined by Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 10
Example of Division of Functions: Let f(x) = x 2 + 2 x and g(x) = 3 x 1 • Find the symbolic representation for the function and use this to evaluate • • So Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 11
Definitions-Composition If f(x) and g(x) both exist, the composition of two functions f and g are defined by Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 12
Composition of Functions-Symbolic Find a symbolic representation for the composite function g ○ f that converts x miles into inches. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 13
Example of Composition of Functions: Let f(x) = x 2 + 2 x and g(x) = 3 x – 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 14
Product and Composition of Two Functions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 15
Evaluating Combinations of Functions Numerically • Given numerical representations for f and g in the table • Evaluate combinations of f and g as specified. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 16
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 17
Evaluating Combinations of Functions Graphically • Use graph of f and g below to evaluate • (f + g) (1) y = f(x) • (f – g) (1) • (f/g) (1) • (f g) (1) y = g(x) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 18
Answers: y = f(x) y = g(x) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 19
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