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One to one functions

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One-to-One Functions

A one-to-one function is a function in which each output comes from exactly one input, so no two inputs share an output. Learn the horizontal line test, why one-to-one functions are the ones that have inverses, and how to check an equation, with examples.

What a one-to-one function is

A one-to-one function is a function in which every output comes from exactly one input. No two different x-values ever map to the same y-value. In other words, a regular function already requires each input to give one output; a one-to-one function adds the reverse rule — each output traces back to a single input. (The formal name is an injective function.) It builds directly on identifying functions.

The quickest way to check is the horizontal line test: if no horizontal line crosses the graph more than once, the function is one-to-one. A straight line passes the test. A parabola fails it, because a horizontal line hits the two arms at two different x-values that share the same y-value.

The horizontal line test Two graphs. On the left a straight line is crossed by a horizontal line only once, so it passes the horizontal line test and is one-to-one. On the right a parabola is crossed by a horizontal line twice, so it fails the test and is not one-to-one. Passes: one-to-one crossed once Fails: not one-to-one crossed twice
The horizontal line test: a line is one-to-one; a parabola is not.

One-to-one is the exact condition a function needs to have an inverse function. If two inputs shared an output, the inverse would not know which input to return, so only one-to-one functions can be reversed. This is why the test shows up so often when you work with inverses.

  • One-to-one: f(x) = 2x + 1 — each output has one input; it passes the horizontal line test.
  • Not one-to-one: f(x) = x² — both x = 2 and x = −2 give 4, so it fails.
  • One-to-one: f(x) = x³ — every output comes from a single input.

To test an equation without a graph, assume f(a) = f(b) and check whether that forces a = b. If it does, the function is one-to-one.

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