Zero of a Function

High School

Definition

Refers to the x-value that makes the function equal to 0. In other words, what input to a function can produce the output of zero? In polynomial functions, the zero is known as the root. The Zero of a function can be a real or a complex number.

Worked examples

\(f(x) = x^2 - 9\); zeros at \(x = 3\) and \(x = -3\) because \(f(3) = 0\) and \(f(-3) = 0\)

Find the x-values that make the output zero; check by substituting them back into the function.

\(g(x) = 2x - 6\); zero at \(x = 3\) because \(2(3) - 6 = 0\)

A linear function has exactly one zero where the line crosses the x-axis.

\(h(x) = x^2 + 1\); no real zeros because \(x^2 + 1\) is always positive

Some functions have no real zeros but may have complex zeros like \(x = i\) and \(x = -i\).

Common mistakes

The zero of \(f(x) = x^2 - 9\) is \(0\) → The zeros are \(x = 3\) and \(x = -3\) A zero is an x-value that makes f(x) equal 0, not the output 0 itself.
\(f(x) = (x - 2)(x + 5)\) has zeros at \(x = -2\) and \(x = 5\) → The zeros are \(x = 2\) and \(x = -5\) Set each factor equal to zero: \(x - 2 = 0\) gives \(x = 2\), not \(x = -2\).
Zeros and y-intercepts are the same thing → Zeros are x-intercepts; the y-intercept is \(f(0)\) Zeros occur where the graph crosses the x-axis, not the y-axis.

Where you'll use it next

Zeros are essential for solving polynomial equations, sketching graphs, analyzing quadratic and higher-degree functions, and tackling calculus problems involving roots and optimization.

Found in 1 StudyPug lesson

Understand relations between x- and y-intercepts

Algebra 1

The relations between x-and y-intercepts can go beyond mathematics. It can represent the various kinds of real-life scenarios. For example, time and distance and money related problems.

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