Inverse Functions.

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5.3 Inverse Function.

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Presentation transcript:

Inverse Functions

Function If for every x there exists at most one y One – to – One Function If for every x there exists at most one y AND for every y there exists at most one x Function Function One – to – One NOT One – to – One

Only one – to – one functions have inverses GRAPHICALLY One – to – One Function Inverse

Function and Inverse have Symmetry about the line y = x One – to – One Function Inverse Function and Inverse have Symmetry about the line y = x

Find the inverse graph of the function below, if it exists. Since not ONE – TO – ONE, no inverse function exists

Find the inverse graph of the function below, if it exists. (4.5, 7) (7, 4.5) (0, 3) (-4, 1) (3, 0) (1, -4) (-7, -5) (-5, -7)

Find the inverse graph of the function below, if it exists.

Finding inverse functions algebraically

Finding inverse functions algebraically PCH ONLY – LOOK AT PROBLEM 68

Derivatives of Inverse Functions If f is differentiable at every point on an interval, and f ’ is never zero on the interval, then: is differentiable at every point on the interior of the Interval and its value at the point f(x) is:

Find the derivative of the inverse of f(x) = 5 – 4x evaluated at c = ½

Find the derivative of the inverse of evaluated at c = 4 Not One-to-One However….One-to-One on [2, 6] which not only includes c = 4, but it also eliminates f ‘ (x) = 0 issue as well Alternative…….

Find the derivative of the inverse of evaluated at c = 4

Find the derivative of the inverse of at c = 1