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Section 7.1

Section 7.1. Inverse Functions. ONE-TO-ONE FUNCTIONS. Definition : A function f is called one-to-one (or 1-1) if it never takes the same value twice; that is, f ( x 1 ) ≠ f ( x 2 ) whenever x 1 ≠ x 2. THE HORIZONTAL LINE TEST.

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Section 7.1

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  1. Section 7.1 Inverse Functions

  2. ONE-TO-ONE FUNCTIONS Definition: A function f is called one-to-one (or 1-1) if it never takes the same value twice; that is, f (x1) ≠ f (x2) whenever x1≠ x2

  3. THE HORIZONTAL LINE TEST A function is one-to-one if and only if no horizontal line intersects the graph more than once.

  4. INVERSE FUNCTION Definition: Let f be a one-to-one function with domain A and range B. Then its inverse functionf−1 has domain B and range A and is defined by f−1(y) = x if, and only if, f (x) = y for any y in B.

  5. COMMENTS ON INVERSE FUNCTIONS • f−1( f (x)) = x for every x in A. • f ( f −1(x)) = x for every x in B. • domain of f −1 = range of f • range of f −1 = domain of f • Do NOT mistake f −1 for an exponent. f −1(x) does NOT mean 1/f (x).

  6. FINDING THE FORMULA FOR AN INVERSE FUNCTION To find the formula for the inverse function of a one-to-one function: STEP 1: Write y = f (x) STEP 2: Solve this equation for x in terms of y (if possible). STEP 3: To express f −1 as a function of x, interchange x and y. The resulting equation is y = f −1(x).

  7. THE GRAPH OF AN INVERSE FUNCTION The graph of f −1 is obtained by reflecting the graph of f about the line y = x.

  8. INVERSE FUNCTIONS AND CONTINUITY Theorem: If f is a one-to-one continuous function defined on an interval, then its inverse function is also continuous.

  9. INVERSE FUNCTIONS AND DIFFERENTIABILITY Theorem: If f is a one-to-one differentiable function with inverse function g = f −1 and f ′(g(a)) ≠ 0, then the inverse function is differentiable at a and Alternatively,

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