One-to-One Function - Maple Programming Help

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One-to-One Function

 Invertible A function $f\left(x\right)$ is invertible if for every point $y$ in the range of $f\left(x\right)$ the equation $y=f\left(x\right)$ can be uniquely solved for $x$.

 One-to-One A function $f\left(x\right)$ is one-to-one if distinct input values are mapped by $f\left(x\right)$ to distinct output values. A synonym for one-to-one is injective.

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.

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