Mathematics · Ch 15 — Functions
Graph of a Function: Vertical and Horizontal Line Tests
Graph of a Function: Vertical and Horizontal Line Tests
Graph of a function. When the domain of a function lies in , the function can be drawn as a curve in the -plane, consisting of every point with .
Vertical Line Test. Given any curve, we can check whether it represents a function of using this test: a graph represents a function of only if no vertical line intersects the curve in more than one point. If every has a unique associated -value (Fig. 6.10), the curve is a function. If some vertical line meets the curve at two or more points (Fig. 6.11), that -value has more than one -value, so the curve is not a function of . …
What this figure shows. A smooth curve in the XY-plane together with one or more sample vertical lines drawn across it, each vertical line shown touching the curve at only a single point. The caption explains that because no vertical line meets the curve more than once, every x-value has a unique associated y-value, so the curve genuinely represents a function o …
What this figure shows. A curve (for instance a sideways-opening shape such as a circle or sideways parabola) together with a sample vertical line that is shown crossing the curve at two separate points. The caption explains that because this one vertical line meets the curve twice, that x-value has two different y-values, so the curve does NOT represent a function …
What this figure shows. Two separate example curves, each shown together with one or more sample horizontal lines, with every horizontal line touching each curve at only a single point. The caption explains that because no horizontal line meets either curve more than once, each function is one-one (injective) — a horizontal line crossing twice would instead mean two different x-values share one y-value. …