Inverse functions reverse the mappings of original functions and are crucial in mathematics. This guide explains how to find an inverse function by interchanging x and y in the original function's equation and solving for the new y. It also covers the importance of bijective functions, the procedure for finding inverses, and the relationship between the domains and ranges of functions and their inverses. Graphical representations and problem-solving with inverse functions are also discussed.
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Inverse functions reverse the correspondence established by an original function
Inverse functions exist exclusively for bijective functions, which are both one-to-one and onto
To find an inverse function, replace f(x) with y, solve for x in terms of y, and interchange x with f⁻¹(x) and y with x
The inverse function of f(x) = 5x + 6 is f⁻¹(x) = (x - 6) / 5
The inverse function of j(x) = x² - 6 is j⁻¹(x) = √(x + 6), with a restricted domain of x ≥ 0
To evaluate an inverse function, substitute the given x value into the inverse function. To solve for x, set the inverse function equal to a known value and solve for x
The domain of an original function becomes the range of its inverse, and the range of the original function becomes the domain of the inverse
For non-one-to-one functions, the domain must be restricted to ensure the existence of an inverse function
To find the range of an inverse function, consider the domain of the original function and calculate the corresponding y values
Graphically, an inverse function is the reflection of the original function across the line y = x
Inverse functions can also be graphed by plotting their points on a graph
It is essential to understand the interdependent relationship between the domains and ranges of functions and their inverses