A function fx is invertible if for every point y in the range of fx the equation y=fx can be uniquely solved for x.
A function fx is one-to-one if distinct input values are mapped by fx to distinct output values. A synonym for one-to-one is injective.
In order for the function fx to be invertible, the problem of solving x=fy 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 fx and the range of the inverse is the domain of fx, this means that in order for fx to be invertible, its graph must satisfy the horizontal line test: Each horizontal line through the graph of y=fx 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.
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