The horizontal line test

For a real function of a real variable, injectivity and surjectivity can be recognised by looking at how horizontal lines meet the graph. Let us compare $y=x^2$ (not injective) and $y=x^3$ (injective): the line $y=3$ intersects the first at two points, the second at just one.

$y=x^2$ is not injective: the line $y=3$ meets it at two points, with abscissae $x=\pm\sqrt{3}\approx\pm 1{,}73$.

$y=x^3$ is injective: the line $y=3$ meets it at a single point, with abscissa $x=\sqrt[3]{3}\approx 1{,}44$.

Remark — Graphical tests

For a real function of a real variable, with graph in the Cartesian plane:

• $f$ is injective $\iff$ every horizontal line intersects the graph at at most one point.
• $f$ is surjective (onto the codomain $\mathbb{R}$) $\iff$ every horizontal line intersects the graph at at least one point.
• $f$ is bijective $\iff$ every horizontal line intersects the graph at exactly one point.

Example

• $f(x)=x^2$ on $\mathbb{R}$ is neither injective ($f(2)=f(-2)=4$) nor surjective onto $\mathbb{R}$ (it does not reach negative values).
• $f(x)=x^3$ on $\mathbb{R}$ is injective and surjective: bijective.
• $f(x)=\tfrac{1}{x}$ from $\mathbb{R}\setminus\{0\}$ to $\mathbb{R}\setminus\{0\}$ is bijective.

Links

Topics: Functions and properties
Concepts: Bijective function · Injective function · Surjective function · Horizontal line test
Functions: Cubic function · Homographic function · Parabola
Skills: Interpreting a graph · Sketching a graph