AP Calculus AB and BC glossary

Root of a function

Also called: Zero of a function, x-intercept

A root of a function is an input where the output equals zero. The same number is called a zero of the function, and on a graph it is an x-intercept. Roots matter because zeros of a derivative locate critical points, and the Intermediate Value Theorem proves a root exists without finding it.

Root, zero, and $x$ -intercept name the same number seen three ways: as a solution of an equation, as an input of a function, and as a point on a graph. In calculus you hunt for roots of derivatives more often than roots of $f$ itself. Solutions of $f'(x) = 0$ are critical points and candidates for extrema, and solutions of $f''(x) = 0$ are candidates for inflection points.

To prove a root exists on $[a, b]$ , state that $f$ is continuous there and show that $f(a)$ and $f(b)$ have opposite signs. Zero then lies between the two endpoint values, so the Intermediate Value Theorem supplies a $c$ in $(a, b)$ with $f(c) = 0$ . For $f(x) = x^3 + x - 1$ , the values $f(0) = -1$ and $f(1) = 1$ pin a root inside $(0, 1)$ .

The mistake

Running the sign-change argument backwards. Opposite signs at the endpoints guarantee a root, but matching signs guarantee nothing. For $f(x) = x^2 - 4$ we get $f(-3) = 5$ and $f(3) = 5$ , the same sign, and yet two roots sit between them at $x = -2$ and $x = 2$ . The Intermediate Value Theorem proves existence and never proves absence.

Appears in: Unit 1: Limits and Continuity