One-to-One Function

In order for the function $f\left(x\right)$ to be invertible, the problem of solving $x\=f\left(y\right)$ for $y$ must have a unique solution. This is because for the inverse to be a function, it must satisfy the property that for every input value in its domain there must be exactly one output value in its range; the inverse must satisfy the vertical line test. Since the domain of the inverse is the range of $f\left(x\right)$ and the range of the inverse is the domain of $f\left(x\right)$, this means that in order for $f\left(x\right)$ to be invertible, its graph must satisfy the horizontal line test: Each horizontal line through the graph of $y\=f\left(x\right)$ must intersect that graph exactly once.

	Creating a One-to-One Function
	Click and drag with your mouse to draw a function in the plot below. The horizontal line test is performed, and the title indicates whether the function you've drawn passes this test (so it is one-to-one). If your function is one-to-one, you can draw its inverse by clicking Invert. Clear an existing graph by clicking it.