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Inverse of Functions bsci-ch.org Topical Outline | Algebra 2 Outline | MathBits" Teacher Resources Terms of Use Contact Person: Donna Roberts
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Inverse functions were examined in Algebra 1. See the Refresher Section to revisit those skills.

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A function and its inverse function can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs some operation on this value, and creates an output answer. The inverse function takes the output answer, performs some operation on it, and arrives back at the original function"s starting value. This "DO" and "UNDO" process can be stated as a composition of functions.


 A function composed with its inverse function yields the original starting value. Think of them as "undoing" one another and leaving you right where you started. If functions f and g are inverse functions,
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Basically speaking, the process of finding an inverse is simply the swapping of the x and y coordinates. This newly formed inverse will be a relation, but may not necessarily be a function.

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The inverse of a function may not always be a function! The original function must be a one-to-one function to guarantee that its inverse will also be a function.
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A function is a one-to-one function if and only if each second element corresponds to one and only one first element. (Each x and y value is used only once.)

Use the horizontal line test to determine if a function is a one-to-one function. If ANY horizontal line intersects your original function in ONLY ONE location, your function will be a one-to-one function and its inverse will also be a function.

The function y = 2x + 1, shown at the right, IS a one-to-one function and its inverse will also be a function.

(Remember that the vertical line test is used to show that a relation is a function.)

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An inverse relation is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function.
If the graph of a function contains a point (a, b), then the graph of the inverse relation of this function contains the point (b, a).

Should the inverse relation of a function f (x) also be a function, this inverse function is denoted by f -1(x).Note: If the original function is a one-to-one function, the inverse will be a function.


If a function is composed with its inverse function, the result is the starting value. Think of it as the function and the inverse undoing one another when composed. Consider the simple function f (x) = {(1,2), (3,4), (5,6)} and its inverse f-1(x) = {(2,1), (4,3), (6,5)}
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More specifically:
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The answer is the starting value of 2.

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Finding inverses: Let"s refresh the 3 methods of finding an inverse.


Swap ordered pairs: If your function is defined as a list of ordered pairs, simply swap the x and y values. Remember, the inverse relation will be a function only if the original function is one-to-one.

Example 1: Given function f, find the inverse relation. Is the inverse relation also a function?

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Answer: Function f is a one-to-one function since the x and y values are used only once.  Since function f is a one-to-one function, the inverse relation is also a function.Therefore, the inverse function is:
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x 1 -2 -1 0 2 3 4 -3
f (x) 2 0 3 -1 1 -2 5 1
Answer: Swap the x and y variables to create the inverse relation. The inverse relation will be the set of ordered pairs:{(2,1), (0,-2), (3,-1), (-1,0), (1,2), (-2,3), (5,4),(1,-3)} Since function f was not a one-to-one function (the y value of 1 was used twice), the inverse relation will NOT be a function (because the x value of 1 now gets mapped to two separate y values which is not possible for functions).