The inverse of a function
The Inverse of a Function
Introduction
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A function is a mathematical relationship between two sets of numbers, often between a set of input values (or the domain) and a set of output values (or the range).
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Inverse of a function reverses the process of the original function. If the original function converts values from set A to set B, the inverse function converts the values from set B back to set A.
Finding the Inverse of a Function
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The process of finding the inverse of a function involves swapping the roles of the input, often represented by
x, and the output, often represented byyorf(x). -
After switching
xandy, the next step involves solving the resulting equation fory. The result is the inverse function, usually represented by the notationf^(-1)(x).
Key Points for the Inverse of a Function
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The graph of an inverse function is a reflection of the graph of the original function in the line
y = x. -
Inverse functions undo the operations of the original function. For instance, if the original function multiplies by 3 and then adds 2, the inverse function would subtract 2 and then divide by 3.
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A function must be one-to-one, meaning each input maps to exactly one output, for an inverse to exist. This is because to reverse the process of a function, there must be one unique action to undo for each input.
Examples
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Given the function
f(x) = 3x + 2, its inverse isf^(-1)(x) = (x - 2) / 3. -
If the function
g(x) = x^2forx >= 0, its inverse isg^(-1)(x) = sqrt(x). -
However, for the function
h(x) = x^2for allx, an inverse function does not exist becauseh(x)is not a one-to-one function; each positive y-value corresponds to two different x-values.
Note
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When considering inverses, it’s important to understand the concept of domain and range. The domain of the inverse function is the range of the original function, and vice versa.
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The composition of a function and its inverse results in the original input; that is,
f(f^(-1)(x)) = xandf^(-1)(f(x)) = x.