Verify that the function f^-1(x) is the inverse of f(x) by...

Verify that the function $f^{-1}(x)$ is the inverse of $f(x)$ by showing that $f\left(f^{-1}(x)\right)=x$ and $f^{-1}(f(x))=x .$ Graph $f(x)$ and $f^{-1}(x)$ on the same axes to show the symmetry about the line $y=x$.
$$f(x)=\frac{1}{x} ; f^{-1}(x)=\frac{1}{x}, x \neq 0$$

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Key Concepts

Inverse Functions

An inverse function reverses the roles of the inputs and outputs of the original function. For a function f(x), its inverse f?¹(x) satisfies both f(f?¹(x)) = x and f?¹(f(x)) = x, meaning that applying the function and its inverse in succession brings you back to the starting value.

Function Composition

Function composition is the process of applying one function to the result of another. Verifying that f(f?¹(x)) = x and f?¹(f(x)) = x confirms that the two functions are inverses, as each composition results in the identity function, which leaves any input unchanged.

Symmetry about the Line y = x

The graphs of a function and its inverse always exhibit symmetry with respect to the line y = x. This geometric property reflects the fact that the roles of the x and y coordinates are interchanged in an inverse function, making the line y = x the axis of reflection.

Domain and Range Restrictions

For a function to have an inverse, it must be bijective, meaning it is both one-to-one and onto. Domain and range restrictions are essential to ensure that the function and its inverse are well-defined and that each input corresponds to exactly one output.

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