6.3 Inverse Functions

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# 6.3 Inverse Functions - PowerPoint PPT Presentation

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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### 6.3 Inverse Functions

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

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...

y = x2

x = y2

The inverse is not a function since it does not pass the vertical line test?

y = x

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.