Section 4.1 Inverse Functions

CONCEPT OF AN INVERSE FUNCTION Idea: An inverse function takes the output of the “original” function and tells from what input it resulted. Note that this really says that the roles of x and y are reversed.

MATHEMATICAL DEFINITION OF INVERSE FUNCTIONS In the language of function notation, two functions f and g are inverses of each other if and only if

NOTATION FOR THE INVERSE FUNCTION We use the notation for the inverse of f(x). NOTE: does NOT mean

ONE-TO-ONE FUNCTIONS A function is one-to-one if for each y-value there is only one x‑value that can be paired with it; that is, each output comes from only one input.

ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS Theorem: A function has an inverse if and only if it is one-to-one.

TESTING FOR A ONE-TO-ONE FUNCTION Horizontal Line Test: A function is one-to-one (and has an inverse) if and only if no horizontal line touches its graph more than once.

GRAPHING AN INVERSE FUNCTION Given the graph of a one-to-one function, the graph of its inverse is obtained by switching x- and y-coordinates. The resulting graph is reflected about the line y = x.

FINDING A FORMULA FOR AN INVERSE FUNCTION To find a formula for the inverse given an equation for a one-to-one function: Replace f (x) with y. 2. Interchange x and y. 3. Solve for y 4. Replace y with f -1(x). Solve the resulting

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