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Combinations & Inverse Functions: Working with Functions and Finding Inverse Functions

Learn how to work with combinations/compositions of functions and find inverse functions. Practice arithmetic combinations and compositions, and understand the definition and graphs of inverse functions.

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Combinations & Inverse Functions: Working with Functions and Finding Inverse Functions

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  1. Combinations of Functions & Inverse FunctionsObj: Be able to work with combinations/compositions of functions. Be able to find inverse functions.TS: Making decisions after reflections and review Warm-Up: Given f(x) = 2x + 3 and g(x) = x2 – 4, find… f(2) + g(2) g(3) – f(3)

  2. Arithmetic Combinations Where the domain is the real numbers that both f and g’s domains have in common. For f/g also g(x) ≠ 0.

  3. Examples Find each of the below combinations given • (f∙g)(2) 3) (f/g)(x) & its domain • (f – g)(2)

  4. Arithmetic Compositions (f○g)(x) = f(g(x)) Given the below find each of the following. • (f ○ g)(2) 2) g(f(-1))

  5. Arithmetic Compositions (f○g)(x) = f(g(x)) Given the below find each of the following. 3) f(g(x)) & its domain

  6. Find f(g(x)) and g(f(x))

  7. Definition of the Inverse of a Function Let f and g be two functions where f(g(x)) = x for every x in the domain of g, and g(f(x)) = xfor every x in the domain off. Under these conditions, g is the inverse of f and g is denoted f-1. Thus f(f-1(x))=x and f-1(f(x))=x where the domain of f must equal the range of f-1 and the range of f must equal the domain of f-1.

  8. Graphs of Inverses Two equations are inverses if their graphs are reflections of one another across the line y=x. y f y=x f -1 1 x 1

  9. Inverse FUNCTIONS A function f(x) has an inverse function if the graph of f(x) passes the ___________________. (In other words the relation is ONE-TO-ONE: For each y there is exactly one x) Circle the functions that are one-to-one (aka have inverse functions) y y y y x x x x y x

  10. Examples: Determine if the two functions f and g are inverses. 1) and

  11. Finding an Inverse • Verify that the function is one-to-one thus has an inverse function using the Horizontal Line Test. • Switch x & y. • Solve for y. • Make sure to use proper inverse notation for y for your final answer. (Ex: f-1(x), not y) • To check your answer: • Verify they are inverses by testing to see if • f(f-1(x)) = f-1(f(x)) = x

  12. f(x) = -.5x + 3 Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the inverse function .

  13. 2) f(x) = x2 – 4 Find the inverse function if there is one, if there is not one, restrict the domain to make it one-to-one then find the inverse function .

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