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Test for 1-1 Function by Horizontal Line Test

A 1-1 function has one unique output for every input. To test for a 1-1 function, verify that any horizontal line cuts the graph of the function only once.

The inverse of a function exist only if the function is a 1-1 function.

Example (not a 1-1 function)

$$ f : x \mapsto x^2 - 2x - 2, x \in \mathbb{R}, -1 < x \le 4 $$

Graphing the function:

Function f.png

There exist horizontal lines, such as $y = -2$, that cuts the graph more than once. Hence, $f$ is not a 1-1 function (and it's inverse function does not exist).

The output of f is not unique

The output of f is not unique

 

Example (1-1 function)

$$ g : x \mapsto x^2 - 2x - 2, \phantom{000} x \in \mathbb{R}, 1 \le x \le 4 $$

Graphing the function:

Restricted g.png

Any horizontal line, $y = k, \phantom{.} -3 \le k \le 6$, cuts the graph only once. Thus, $g$ is a 1-1 function (and it's inverse function exists).