Inverse Functions. DEFINITION Two relations are inverses if and only if when one relation contains (a,b), the other relation contains (b,a).

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Presentation transcript:

Inverse Functions

DEFINITION Two relations are inverses if and only if when one relation contains (a,b), the other relation contains (b,a).

EXAMPLES Suppose and. Find:

EXAMPLES Suppose and. Find:

IMPORTANT IDEA Did you notice that : and In general: and

EXAMPLE Let The inverse of f(x) is: The graph of f(x) and f -1 (x) looks like…

IMPORTANT IDEA The last example leads us to: f -1 (x) is a reflection of f(x) over the line y = x.

FINDING AN INVERSE Given a function, find its inverse by: 1. Letting y = f(x) 2. Interchange x and y 3. Solve for the new y 4. Rename y as f -1 (x) {only if it is a function)

EXAMPLE Find the inverse of the following functions: 1. 2.

EXAMPLE Find the inverse of the following functions: 1. 2.

EXAMPLES CONTINUED 3.4.

EXAMPLES CONTINUED 3.4.

IMPORTANT IDEA The inverse of a function may or may not be a function. For example:

HORIZONTAL LINE TEST If any horizontal line intersects the graph of a function in no more than one point, its inverse is a function.

DEFINITION If the original and the inverse are both functions, then it is said to be one-to-one. No x-values AND no y-values repeat!

ASSIGNMENT RED BOOK Page Problems 9-21odd, 31-39odd, and 47-53odd