4 2 operations on functions
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This document discusses operations on functions including adding, subtracting, multiplying, and dividing functions. It also covers composing functions by applying one function to the output of another. The key objectives are to perform these operations on functions and determine the resulting domains. Examples are provided to demonstrate finding the sum and difference of functions, as well as composing functions and determining the domains of composite functions.
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4 2 operations on functions
- 1. 4-2 OPERATIONS ON FUNCTIONS Objectives: 1. Add, subtract multiply and divide functions. 2. Compose functions 3. Determine the domains of resulting functions
- 2. +, -, ×, ÷ FUNCTIONS Sum: (f + g)(x) = f(x) + g(x) Difference: (f – g)(x) = f(x) – g(x) Product: (f ∙ g)(x) = f(x) ∙ g(x) Quotient:
- 3. EXAMPLE 1 Let f(x) = x + 1 and g(x) = x2 – 1, find: (f + g)(x)
- 4. YOU TRY! Let f(x) = x2 + x and g(x) = x + 1, find: (f – g)(x) (f ∙ g)(x)
- 5. ADDING/SUBTRACTING GRAPHS To find (f + g)(x) or (f – g)(x) from their graphs, add or subtract the corresponding y-values. Example: f(-2) – g(-2) f(-4) + g(-4) f(6) – g(6) f(x) g(x)
- 6. YOU TRY! Find: (f + g)(-6) (f – g)(-2) (g – f)(1) (g + f)(5) f(x) g(x)
- 7. FUNCTION COMPOSITION One function becomes the input of another function (f ◦ g)(x) = f(g(x)) g(x) becomes the input for f (g ◦ f)(x) = g(f(x)) f(x) becomes the input for g
- 8. COMPOSITE FUNCTION DOMAINS The domain of f ◦ g includes x values such that: 1. x is in the domain of g 2. g(x) is in the domain of f So: Find domain of input function Modify (if needed) so it also works for the new composite function
- 9. EXAMPLE 2 Let and Find a rule for (f ◦ g)(x) and give the domain.
- 10. EXAMPLE 3 Let and Find a rule for (f ◦ g)(x) and (g ◦ f)(x) and give the domain of each.
- 11. YOU TRY! Let and Find (f ◦ g)(x) and its domain.
- 12. EXTRA EXAMPLE Let , and Find h(g(f(4)) Find h(g(f(x))
- 13. YOU TRY! Let , and Find g(h(f(9)) Find g(h(f(x))