An inverse is a mathematical operation that reverses the effect of another operation. It is a fundamental concept in functions, where the inverse function undoes the original function, restoring the original input value.
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The inverse of a function reverses the input and output values, effectively undoing the original function.
For a function to have an inverse, it must be one-to-one, meaning each output value is paired with only one input value.
The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.
Inverse functions can be used to solve equations by isolating the variable of interest.
Graphically, the inverse of a function is the reflection of the original function across the line $y = x$.
Review Questions
Explain how the inverse of a function is related to the original function.
The inverse of a function reverses the input and output values, effectively undoing the original function. This means that if $f(x) = y$, then the inverse function $f^{-1}(y) = x$. The domain of the inverse function becomes the range of the original function, and the range of the inverse function becomes the domain of the original function. This relationship allows inverse functions to be used to solve equations by isolating the variable of interest.
Describe the conditions required for a function to have an inverse.
For a function to have an inverse, it must be one-to-one, meaning each output value is paired with only one input value. This ensures that the inverse function can uniquely determine the original input value from the output value. Graphically, this means that the function must pass the horizontal line test, where no horizontal line intersects the graph of the function more than once. If a function is not one-to-one, it may still have an inverse, but the inverse will not be a function.
Explain how the graph of an inverse function is related to the graph of the original function.
The graph of the inverse function is the reflection of the original function across the line $y = x$. This means that if the original function $f(x)$ has a point $(a, b)$ on its graph, then the inverse function $f^{-1}(x)$ will have the point $(b, a)$ on its graph. This relationship allows us to easily sketch the graph of an inverse function if we know the graph of the original function, and vice versa.