Lesson 50: Finding Inverses of Relations and Functions (Exploration: Graphing a Function and its Inverse)

New Concept

If $r$ represents a relation, then the inverse relation is the set of ordered pairs obtained by reversing the coordinates in each ordered pair of $r$.

What’s next

Next, you'll put this concept into practice by swapping variables to find the inverse of different equations and relations.

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.

To find an equation for the inverse of a function like $y = 2x + 6$, first interchange the variables (swap $x$ and $y$). Then, solve the resulting equation for $y$ to get the inverse function's formula.

Find the inverse of $y = 5x + 10$. Swap variables to get $x = 5y + 10$. Solve for y: $x - 10 = 5y$, so $y = \frac{1}{5}x - 2$.: The inverse of $y = -3x + 9$ is found by swapping to $x = -3y + 9$. Solving gives $x-9 = -3y$, so $y = -\frac{1}{3}x + 3$.

Think of this as a two-step mission to uncover a function's secret identity. First, you pull the ultimate switcheroo: $x$ becomes $y$ and $y$ becomes $x$. Then, you unleash your algebra skills to solve for the new $y$, isolating it on one side. Mission accomplished! You have successfully found the formula for the inverse function.

Vertical Line Test: A relation is a function if and only if no vertical line intersects its graph in more than one point.
Horizontal Line Test: A relation's inverse is a function if and only if no horizontal line intersects the graph of the original relation in more than one point.

The graph of $y=x^2$ is a parabola. It passes the vertical line test (it is a function) but fails the horizontal line test (its inverse is not a function).
A straight line like $y=4x-1$ passes both tests, so both the line and its inverse are functions.
A circle's graph fails both tests, meaning neither the relation nor its inverse is a function.

Use imaginary laser beams to test your graphs! The vertical line test scans up and down. If a vertical line ever hits your graph in more than one spot, it's not a function. The horizontal line test scans left and right on the original graph. If a horizontal line hits more than once, its inverse will not be a function.