Zero of a Function — Definition, Graph & Examples

Zero of a Function

A value of x which makes a function f(x) equal 0. A zero may be real or complex.

Example: 3 is a zero of f(x) = x² - 4x + 3, because f(3) = 3² - 4(3) + 3 = 0.

See also

Key Formula

f(x) = 0

Where:

$f(x)$ = The function being evaluated
$x$ = The input value; any value of x satisfying this equation is called a zero of f

Worked Example

Problem: Find the zeros of f(x) = x² − 5x + 6.

Step 1: Set the function equal to zero.

x^2 - 5x + 6 = 0

Step 2: Factor the quadratic expression. Look for two numbers that multiply to 6 and add to −5. Those numbers are −2 and −3.

(x - 2)(x - 3) = 0

Step 3: Apply the zero-product property: set each factor equal to zero.

x - 2 = 0 \quad \text{or} \quad x - 3 = 0

Step 4: Solve each equation.

x = 2 \quad \text{or} \quad x = 3

Step 5: Verify: f(2) = 4 − 10 + 6 = 0 and f(3) = 9 − 15 + 6 = 0. Both values check out.

f(2) = 0, \quad f(3) = 0

Answer: The zeros of f(x) = x² − 5x + 6 are x = 2 and x = 3.

Another Example

This example shows that zeros can be complex numbers. Unlike the first example, there are no real zeros here, which means the parabola sits entirely above the x-axis.

Problem: Find all zeros (including complex zeros) of f(x) = x² + 4.

Step 1: Set the function equal to zero.

x^2 + 4 = 0

Step 2: Isolate x².

x^2 = -4

Step 3: Take the square root of both sides. Since the right side is negative, the solutions involve the imaginary unit i, where i² = −1.

x = \pm\sqrt{-4} = \pm 2i

Step 4: Verify: f(2i) = (2i)² + 4 = −4 + 4 = 0. Confirmed.

f(2i) = 0, \quad f(-2i) = 0

Answer: The zeros are x = 2i and x = −2i. These are complex zeros, so the graph of f(x) = x² + 4 never touches the x-axis.

Frequently Asked Questions

What is the difference between a zero and a root of a function?

In most algebra courses, 'zero' and 'root' mean the same thing — a value of x where f(x) = 0. Some textbooks reserve 'root' for solutions of an equation (like x² − 9 = 0) and 'zero' for values that make a function equal zero (like the zeros of f(x) = x² − 9). In practice, the two terms are used interchangeably.

Are zeros the same as x-intercepts?

Real zeros and x-intercepts are closely related but not identical concepts. Every real zero of a function gives you an x-intercept on its graph, located at the point (x, 0). However, complex zeros do not appear on the real-number graph at all, so they are not x-intercepts. When someone says 'x-intercept,' they always mean a real number.

How many zeros can a polynomial function have?

By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n zeros when you count complex zeros and repeated zeros (multiplicities). For example, a cubic polynomial (degree 3) always has exactly 3 zeros total, though some may be complex or repeated.

Zero of a Function vs. x-Intercept

	Zero of a Function	x-Intercept
Definition	A value x where f(x) = 0	A point (x, 0) where the graph crosses the x-axis
Type of object	A number (real or complex)	A coordinate point on the graph
Includes complex values?	Yes — zeros can be complex numbers	No — x-intercepts are always real
Notation	x = 3 is a zero of f	(3, 0) is an x-intercept

Why It Matters

Finding zeros is one of the most fundamental tasks in algebra and precalculus. You use it to solve equations, factor polynomials, and analyze where a graph crosses the x-axis. Zeros also appear throughout science and engineering — for instance, finding when a projectile hits the ground means finding the zero of a height function.

Common Mistakes

Mistake: Confusing the zero of a function with the value of the function. Students sometimes say "the zero is f(x) = 0" instead of identifying the x-value.

Correction: The zero is the input x that produces an output of 0. If f(3) = 0, then the zero is x = 3, not "f(x) = 0."

Mistake: Assuming every zero appears on the graph as an x-intercept.

Correction: Only real zeros show up as x-intercepts. Complex zeros (like x = 2i) are valid zeros of the function but do not correspond to any point on the real-number graph.

Related Terms

Function — A zero is defined in terms of a function
Root — Synonym for zero of a function
x-Intercept — Graph point corresponding to a real zero
Real Numbers — Real zeros belong to this number set
Complex Numbers — Zeros may be complex numbers
Factor Theorem — Links zeros to factors of a polynomial
Quadratic Formula — Key method for finding zeros of quadratics
Fundamental Theorem of Algebra — Guarantees n zeros for a degree-n polynomial