3.6 - Inverse Functions Notation: Say: “f-inverse of x”… Definition: “For 2 functions f and g, if and then function g is the inverse of function f and vice versa”. ex: Finding informally: Verifying algebraically ex:

Verifying algebraically ex: Graphs of Inverses If point (a, b) is on f, then point (b, a) is on f -1 (and vice-versa) The graph of f -1 is a reflection of the graph of f across the line y = x ex:

f(a) = 3a-2 f(b) = 3b-2 [Set f(a) = f(b)] f(a) = a2 - a 3a-2 = 3b-2 One-to-One Function A function for which no two elements of the domain of the function correspond to one element of the range. One-to-one is often written 1-1. Note: y = f(x) is a function if it passes the vertical line test. It is a 1-1 function if it passes both the vertical line test and the horizontal line test. Another way of testing whether a function is 1-1 is given below: Test for 1-1 functions: “If f(a) = f(b) implies that a = b, then f is 1-1.” Is f(x) = 3x-2 one-to-one? f(a) = 3a-2 f(b) = 3b-2 [Set f(a) = f(b)] 3a-2 = 3b-2 3a = 3b a = b implied Is f(x) = x2 - x one-to-one? f(a) = a2 - a f(b) = b2 - b [Set f(a) = f(b)] a2 - a = b2 – b a = b not implied ex: 22 - 2 = (-1)2 - (-1)

x3 x2 Horizontal line test Every horizontal line passes through the graph at only one point… the function is 1-1 x2 A horizontal line passes through the graph at more than one point… the function is not 1-1…

Note: A function that is increasing or decreasing over its entire domain is automatically one-to-one… Finding the Inverse of a Function Steps: Replace f(x) with y. Switch x and y. If the new equation is not a function, then f does not have an inverse. If the new equation is a function, then solve the new equation for y. Replace y with f -1. Verify.

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