Zero of a Function¶

Definition¶

A zero of a function (also called a root or $x$-intercept) is any input value that makes the function's output equal to zero.

Interpretation¶

A zero of $f$ is an input at which the output is exactly $0$, so finding the zeros means solving the equation $f(x)=0$$0$. Graphically, a zero is where the function's graph crosses or touches the $x$-axis.

The zeros of $f$ are the $x$-intercepts of its graph: $f(x_1)=f(x_2)=f(x_3)=0$ for $f(x)=x^3-3x+1$$1$. These zeros, approximately $-1.88$, $0.35$ and $1.53$, are not easy to find by hand. The values $f(-2)=-1$$-1$, $f(-1)=3$$3$, $f(0)=1$$1$, $f(1)=-1$ and $f(2)=3$ change sign on $(-2,-1)$, on $(0,1)$ and on $(1,2)$, and the figure shows the graph crossing the axis once in each of these intervals.

A zero is an element of the domain, a number $x$, not a point of the plane: the zeros of $x^2-1$ are the numbers $1$ and $-1$, while the graph meets the axis at the points $(1,0)$ and $(-1,0)$. Solving $f(x)=g(x)$ is the same as finding the zeros of $f-g$$g$, so every equation in one unknown can be written as a zero problem.

A common slip concerns quotients. A zero of the denominator is not a zero of the function: $1/x$ is not defined at $0$, and it has no zeros at all, since $1/x=0$ has no solution. Later, the intermediate value theorem turns the sign changes above into a proof that a zero exists.

Key Results¶

• A zero of a function is a solution to the equation $f(x)=0\text{.}$

Examples¶

• The zeros of the quadratic polynomial $f(x)=x^2-1$ are $x=1$ and $x=-1\text{.}$
• The function $f(x)=2^x$ does not have any zeros.

Prerequisites¶

• Function

Related Concepts¶

• Graph of a Function
• Singularity
• Fundamental Theorem of Algebra
• Stationary Point

Sources¶

• Math I: Calculus · Lecture 4

Aliases: root of a function, x-intercept Keywords: root

Last updated: 20 October 2026 03:02