Given any function, f, the inverse of the function, f -1 , is a relation that is formed by interchanging each (x,y) of f to a (y,x) of f -1 .
Introduction
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Let f be defined as the set of values given by Let f -1 be defined as the set of values given by 10 -5 4 0 y-values 7 4 0 -2 x-values 7 4 0 -2 y-values 10 -5 4 0 x-values
For a function to have an inverse that is a function, then the original function must be a 1:1 function. A 1:1 function is defined as a function in which each x is paired with exactly one y and y is paired with exactly one x. To be a function we can have no domain value paired with two range values .
That is, the function could not have (-3,5) and (-3,-3) as two points .
To be a function the graph of the function must pass the “vertical line” test. To be a 1:1 function the graph of the function must also pass a “horizontal line” test.
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Example: Inverse Relation Algebraically Example1 : Find the inverse relation algebraically for the function f ( x ) = 3 x + 2. y = 3 x + 2 Original equation defining f x = 3 y + 2 Switch x and y . 3 y + 2 = x Reverse sides of the equation. To calculate a value for the inverse of f , subtract 2, then divide by 3 . To find the inverse of a relation algebraically , interchange x and y and solve for y . y -1 = Solve for y.
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y = x The graphs of a relation and its inverse are reflections in the line y = x . The ordered pairs of f a re given by the equation . Example 1a : Find the graph of the inverse relation geometrically from the graph of f ( x ) = x y 2 -2 -2 2 The ordered pairs of the inverse are given by .
Let f (x)= y = 3x , find the inverse. This is, f -1 : x = 3y .
5x - 1 5y - 1
Solve for y: x (5y – 1) = 3y.
5xy - x = 3y
5xy – 3y = x
y(5x – 3) = x
y = x
5x - 3
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Example 3: Find the inverse of f(x) = f -1 (x) = -2x +2 (-2) (-2) Replace f(x) with y. Interchange x and y. Solve for y. Replace y with f-1(x).
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Example 4: Two functions f and g are inverse functions if and only if both of their compositions are the identity function; f(x) = x. Determine whether and are inverse functions. You must do [f ◦ g](x) and [g ◦ f ](x), if they both equal x, they are inverses!
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[f ◦ g](x) = x + 6 – 6 = x [g ◦ f ](x) = x – 8 + 8 = x So, they ARE inverses of each other!
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Example: Composition of Functions It follows that g = f -1 . Example 5 :Verify that the function g ( x ) = is the inverse of f ( x ) = 2 x – 1. f( g ( x ) ) = 2 g ( x ) – 1 = 2( ) – 1 = ( x + 1) – 1 = x g ( f ( x ) ) = = = = x
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