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Inverse Functions and their Representations

Inverse Functions and their Representations. Lesson 5.2. Definition. A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) } But ... what if we reverse the order of the pairs?

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Inverse Functions and their Representations

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  1. Inverse Functions and their Representations Lesson 5.2

  2. Definition • A function is a set of ordered pairs with no two first elements alike. • f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) } • But ... what if we reverse the order of the pairs? • This is also a function ... it is the inverse function • f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

  3. Example • Consider an element of an electrical circuit which increases its resistance as a function of temperature. R = f(T)

  4. Example • We could also take the view that we wish to determine T, temperature as a function of R, resistance. T = g(R) Now we would say that g(R) and f(T) are inverse functions

  5. Terminology • If R = f(T) ... resistance is a function of temperature, • Then T = f-1(R) ... temperature is the inverse function of resistance. • f-1(R) is read "f-inverse of R“ • is not an exponent • it does not mean reciprocal

  6. Does This Have An Inverse? • Given the function at the right • Can it have an inverse? • Why or Why Not? • NO … when we reverse the ordered pairs, the result is Not a function • We would say the function is not one-to-one • A function is one-to-onewhen different inputs always result in different outputs

  7. One-to-One Functions • When different inputs produce the same output • Then an inverse of the function does not exist • When different inputs produce different outputs • Then the function is said to be “one-to-one” • Every one-to-one function has an inverse • Contrast

  8. One-to-One Functions • Examples • Horizontal line test?

  9. Try Finding the Inverse

  10. Composition of Inverse Functions • Consider • f(3) = 27   and   f -1(27) = 3 • Thus, f(f -1(27)) = 27 • and f -1(f(3)) = 3 • In general   f(f -1(n)) = n   and f -1(f(n)) = n(assuming both f and f -1 are defined for n)

  11. Graphs of Inverses • Again, consider • Set your calculator for the functions shown Dotted style Use Standard Zoom Then use Square Zoom

  12. Graphs of Inverses • Note the two graphs are symmetric about the line y = x

  13. Investigating Inverse Functions • Consider • Demonstrate that these are inverse functions • What happens with   f(g(x))? • What happens with  g(f(x))? Define these functions on your calculator and try them out

  14. Domain and Range • The domain of f is the range of f -1 • The range of f is the domain of f -1 • Thus ... we may be required to restrict the domain of f so that f -1 is a function

  15. Domain and Range • Consider the function    h(x) = x2 - 9 • Determine the inverse function • Problem =>  f -1(x) is not a function

  16. Assignment • Lesson 5.2 • Page 396 • Exercises 1 – 93 EOO

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