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 $y=x$
$$f(x)=2 x+1 ; f^{-1}(x)=\frac{x-1}{2}$$

Key Concepts

Inverse Functions

An inverse function reverses the operation of the original function. When you have a function f(x), its inverse f?¹(x) is defined such that applying f followed by f?¹ (or vice versa) will return the original input. This relationship means that f(f?¹(x)) = x and f?¹(f(x)) = x, which are key properties used to verify that one function is truly the inverse of the other.

Function Composition

Function composition involves applying one function to the result of another. In the context of inverse functions, the composition f(f?¹(x)) or f?¹(f(x)) is used to test and demonstrate that the functions are inverses because the compositions yield the identity function. This confirms that each function undoes the effect of the other.

Graphical Representation of Inverses

Graphing a function and its inverse on the same coordinate system provides a visual check for their relationship. The inverse function is usually reflected across the line y = x, which means that for every point (a, b) on the graph of the function, there is a corresponding point (b, a) on the graph of its inverse.

Symmetry About the Line y = x

The line y = x acts as a mirror line for a function and its inverse. If a function has an inverse, their graphs will be symmetric with respect to y = x. This symmetry is a visual confirmation of the inverse relationship, as swapping the x and y coordinates of a point on one graph will yield a point on the inverse function's graph.

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