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6.3 Inverse Functions

6.3 Inverse Functions. ©2001 by R. Villar All Rights Reserved. Inverse Functions. An inverse of a relation (set of ordered pairs) is obtained by switching the x and y in the ordered pairs. For example, the inverse of {(0, –3), (2, 1), (6, 3)} is: {(–3, 0), (1, 2), (3, 6)}

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6.3 Inverse Functions

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  1. 6.3 Inverse Functions ©2001 by R. Villar All Rights Reserved

  2. Inverse Functions An inverse of a relation (set of ordered pairs) is obtained by switching the x and y in the ordered pairs. For example, the inverse of{(0, –3), (2, 1), (6, 3)} is:{(–3, 0), (1, 2), (3, 6)} The graph of the inverse is the reflection of the graph of the original relation. The mirror of the reflection is the line y = x. inverse original relation “mirror” y = x

  3. Example: What is the inverse of y = 3x – 2? Switch the x and the y. The inverse isx = 3y – 2 Inverses of functions are found the same way…however, the inverse of a function may or may not be a function... For example: Find the inverse off(x) = x2. Is the inverse a function? Replace f(x) with y y = x2 The inverse isx = y2 Graph each by making a table of values...

  4. Original function: f(x) = x2Inverse: x = y2 y = x2 x = y2 The inverse is not a function since it does not pass the vertical line test? y = x

  5. Composition can be used to verify if two functions are inverses of each other... If the functions f and g are inverses of each other, thenf(g(x)) = xand g(f(x)) = x Example: Verify that the functions below are inverses f(x) = 2x – 1g(x) = 1/2 x + 1/2 f(g(x)) = f(1/2 x + 1/2) = 2(1/2 x + 1/2) – 1 = x + 1 – 1 = x g(f(x)) = g(2x – 1) = 1/2( 2x – 1) + 1/2 = x – 1/2 + 1/2 = x Therefore, f and g are inverses of each other.

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