ACT® · Math: Functions — Composition, Transformation, and Inverse Functions
Inverse Graphs as a Reflection Across y = x
The graph of an inverse function is a mirror image of the original graph, flipped across the line y = x. Every point (a, b) on f becomes the point (b, a) on its inverse.
A function takes an input and produces an output. Its inverse undoes that: it takes the output back to the input. On a graph, that swap of roles shows up as a swap of coordinates. If (3, 8) is on f, it means f sent 3 to 8. The inverse sends 8 back to 3, so (8, 3) is on the inverse. Swapping the x and y coordinates of every point is exactly what reflecting across the line y = x does, since that line is where x and y trade places. Fold the paper along y = x and every point lands on its swapped twin.
To graph an inverse without solving for it algebraically, pick a few easy points off the original curve, swap each pair of coordinates, and plot the results. For f(x) = 2x + 1, the points (0, 1), (1, 3), and (2, 5) become (1, 0), (3, 1), and (5, 2) on the inverse, and those three points sit on the line you get from solving y = 2x + 1 for x.
This reflection also explains why not every curve has an inverse that is a function. The horizontal line test on the original becomes the vertical line test on the reflection, since reflecting swaps horizontal and vertical. A parabola fails the horizontal line test, so its reflection fails the vertical line test and isn't a function on its own. This is also why 'inverse' never means '1 divided by f' — that's a different operation, called the reciprocal, and it doesn't undo f's input-output pairing at all.
Key terms
- Inverse function
- A function that reverses another function's input-output pairs, sending each output back to its original input.
- Reflection across y = x
- A transformation that swaps the x and y coordinate of every point on a graph.
- Horizontal line test
- A check where a horizontal line crossing a graph more than once means the graph's inverse won't be a function.
- Vertical line test
- A check where a vertical line crossing a graph more than once means the graph itself isn't a function.
Watch it explained
- How it worksHow does the point (3, 8) on a curve guarantee the point (8, 3) on its inverse?
- Worked exampleDraw the inverse of f(x) = 2x + 1 by reflecting three of its points across y = x.
- Common mistakeWhy isn't the inverse of a function the same as one divided by that function?
- Common mistakeWhy doesn't every curve you can draw have an inverse that is itself a function?
Check yourself
End of topic.