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MCS121 Calculus I Section 0.4 Inverse Functions

MCS121 Calculus I Section 0.4 Inverse Functions. Definition of Inverse Functions. Definition 0.4.1 (p. 39). Example. Show that and are inverses. How to Determine Inverse Functions. A Procedure for Finding the Inverse of a Function f (p. 40).

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MCS121 Calculus I Section 0.4 Inverse Functions

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  1. MCS121Calculus I Section 0.4 Inverse Functions

  2. Definition of Inverse Functions Definition 0.4.1 (p. 39)

  3. Example Show that and are inverses.

  4. How to Determine Inverse Functions A Procedure for Finding the Inverse of a Function f (p. 40)

  5. Example Find a formula for if

  6. Domain and Range Domain of f -1(x) = Range of f(x) Range of f -1(x) = Domain of f(x) Why? If x is in the domain of f then y = f(x) is in the range. But, x = f -1(y), so y is in the domain of f -1 and x is in the range.

  7. Reflection Property of Inverses Figure 0.4.9 (p. 43)

  8. Reflection Property of Inverses Theorem 0.4.5 (p. 43)

  9. Existence of Inverses Suppose f(x) has an inverse. Then, for any two different x’s, say x1 ≠ x2 ,we must have f(x1) ≠ f(x2). Why?? Definition: We say that f(x) is one-to-one if f(x1) ≠ f(x2) whenever x1 ≠ x2 .

  10. Existence of Inverses: One-to-one Figure 0.4.3 (p. 42)

  11. Existence of Inverses Theorem: A function f(x) has an inverse if and only if it is one-to-one. Example: Which of the following are One-to-one (and thus invertible). f(x) = x2 f(x) = x3 + 2 f(x) = sin(x)

  12. Existence of Inverses Theorem 0.4.4 (p.42)The Horizontal Line Test

  13. More Practice

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