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One-to-One and Inverse Functions Section 3.5

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Function and One-to-One Function- each value of x corresponds to only one y – value Use vertical line test on graph One-to-One Function- each x value corresponds to one y value (it’s a function) and no two x values correspond to the same y value Use the Horizontal Line Test If every horizontal line intersects the function at most in one place, then it is one-to-one

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Decide if the relation is a function. If it is a function decide if it is one-to-one. xy xy xy Not a function Function, but not one-to-one Function and one-to-one

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Decide if the relation is a function and one-to-one Not a function (Doesn’t pass vertical line test) Function and one- to-one (Passes vertical and horizontal line test) Function, but not one-to-one (Passes vertical, but not horizontal line test)

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Inverse Function If a function is one-to-one, then it is possible to map each y value back to the x value. f(x) is the original function f - ¹(x) is the inverse function (read “f inverse”) To verify two functions are inverses of each other: f(f - ¹(x)) = x and f - ¹(f(x)) = x

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Verify that f(f -1 (x)) = x and f -1 (f(x))= x

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Verify that f(f ¹(x)) = x and f ¹(f(x)) = x

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Graphical Interpretation of Inverse Functions The graph of the inverse function will be the reflection across the line y = x Each (x,y) on the function will be the point (y,x) on the inverse

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Graph the inverse of the one-to-one function given Select points on the graph (-5,4) (0,3) The inverse function has the Points (4,-5) (3,0)

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Finding the inverse of a function To find the inverse of the function 1. Rewrite the equation with y in place of f(x) 2. Switch the y and the x 3. Solve for y 4. Replace y with f ¹(x)

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Find the inverse of f(x) = x - 9 f(x) = x – 9 y = x – 9 x = y – 9 x + 9 = y

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Find the inverse of the the function f(x) = ½x³ (The function is one-to-one) f(x) = ½x³ y = ½x³ x = ½y³ 2x = y³

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