4-5 Inverses of Linear Functions

The inverse relation is the set of ordered pairs obtained by reversing the coordinates in each ordered pair of a relation $r$. So if $(a, b)$ is in relation $r$, then $(b, a)$ is in the inverse relation. The inverse may or may not be a function.

The inverse of the relation {(-4, 8), (0, 2), (3, 2)} is {(8, -4), (2, 0), (2, 3)}.: If a function's graph contains the point $(-3, 9)$, its inverse relation must contain the point $(9, -3)$.: Graphically, a point $(a, b)$ and its inverse $(b, a)$ are perfect reflections of each other across the line $y=x$.

Imagine your coordinates are wearing shoes on the wrong feet! To find the inverse, you just swap them. The x-value becomes the y-value, and the y-value becomes the x-value. If a point is $(5, 1)$, its inverse buddy is $(1, 5)$. This simple switcheroo gives you the inverse for every single point in the relation.

Property

The notation $f^{-1}(x)$ means the inverse function of $f$, NOT the reciprocal of $f(x)$.

Property

To graph an inverse function $f^{-1}(x)$, reflect the graph of $f(x)$ across the line $y = x$ by swapping the $x$ and $y$ coordinates of each point. If $(a, b)$ is on the graph of $f(x)$, then $(b, a)$ is on the graph of $f^{-1}(x)$.

Examples