Zero of a function

Key takeaways

Zero of a function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Zero of a function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Zero of a function from memory before moving on to harder problems.

Reference excerpt

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain of f {\displaystyle f} such that f ( x ) {\displaystyle f(x)} vanishes at x {\displaystyle x} ; that is, the function f {\displaystyle f} attains the value of 0 at x {\displaystyle x} , or equivalently, x {\displaystyle x} is a solution to the equation f ( x ) = 0 {\displaystyle f(x)=0} . A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f {\displaystyle f} of degree two, defined by f ( x ) = x 2 − 5 x + 6 = ( x − 2 ) ( x − 3 ) {\displaystyle f(x)=x^{2}-5x+6=(x-2)(x-3)} has the two roots (or zeros) that are 2 and 3.

f ( 2 ) = 2 2 − 5 × 2 + 6 = 0 and f ( 3 ) = 3 2 − 5 × 3 + 6 = 0. {\displaystyle f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.}

If the function maps real numbers to real numbers, then its zeros are the x {\displaystyle x} -coordinates of the points where its graph meets the x-axis. An alternative name for such a point ( x , 0 ) {\displaystyle (x,0)} in this context is an x {\displaystyle x} -intercept.

Solution of an equation Every equation in the unknown x {\displaystyle x} may be rewritten as

f ( x ) = 0 {\displaystyle f(x)=0}

by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function f {\displaystyle f} . In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.

Polynomial roots

Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).

Fundamental theorem of algebra

The fundamental theorem of algebra states that every polynomial of degree n {\displaystyle n} has n {\displaystyle n} complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

Computing roots

There are many methods for computing accurate approximations of roots of functions, the best being Newton's method, see Root-finding algorithm. For polynomials, there are specialized algorithms that are more efficient and may provide all roots or all real roots; see Polynomial root-finding and Real-root isolation. Some polynomial, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients; see Solution in radicals.

Zero set

In various areas of mathematics, the zero set of a function is the set of all its zeros. More precisely, if f : X → R {\displaystyle f:X\to \mathbb {R} } is a real-valued function (or, more generally, a function taking values in some additive group), its zero set is f − 1 ( 0 ) {\displaystyle f^{-1}(0)} , the inverse image of { 0 } {\displaystyle \{0\}} in X {\displaystyle X} . Under the same hypothesis on the codomain of the function, a level set of a function f {\displaystyle f} is the zero set of the function f − c {\displaystyle f-c} for some c {\displaystyle c} in the codomain of f . {\displaystyle f.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Zero of a function

Start with the simplest possible case. Write down what Zero of a function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Zero of a function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Zero of a function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Zero of a function

In research: Zero of a function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Zero of a function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Zero of a function is common in secondary-school and first-year university syllabi. It links to neighbouring topics 0 (number), Elementary mathematics, Functions and mappings, so understanding it makes those chapters shorter.
In everyday life: Look for Zero of a function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Zero of a function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Zero of a function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Zero of a function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Zero of a function in simple terms?

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain of f {\displaystyle f} such that f ( x ) {\displaystyle f(x)} vanishes at x {\displaystyle x} ; that is, the function f…

Why does Zero of a function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Zero of a function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Zero of a function.