Zeros Of A Function - Definition, Formula, Graph, Examples - Cuemath

Zeros of a Function

The zeros of a function are the values of the variable of the function such that the values satisfy the equation and give the value of the function equal to 0. Graphically, we can understand the zeros of a function as the x-coordinates (x-intercepts) where the graph cuts the x-axis. We can determine if the zeros of a quadratic function are real, complex, or repeated using the discriminant formula. So, for a function f(x), its zeros are values of x when f(x) = 0. Hence, if we have f(a) = 0, then 'a' is a zero of the function f(x).

In this article, we will explore the concept of zeros of a function and its formula to find the zeros. We will also learn to find the zeros graphically, zeros of a polynomial, and quadratic function with the help of solved examples for a better understanding of the concept.

1.	What are Zeros of a Function?
2.	Zeros of a Function Formula
3.	Finding Real Zeros of a Function
4.	Zeros of a Function on a Graph
5.	FAQs on Zeros of a Function

What are Zeros of a Function?

The zeros of a function f(x) are values of the variable x such that the values satisfy the equation f(x) = 0. The zeros of a function are also called the roots of a function. We can find these zeros graphically as well by determining the x-intercepts of the graph. To determine the type of zeros of a quadratic function algebraically, we use the discriminant formula. To find the zeros of a function in general, we can factorize the function using different methods. Let us understand the meaning of the zeros of a function given below.

Zeros of a Function Definition

The zeros of a function are defined as the values of the variable of the function such that the function equals 0. Graphically, the zeros of a function are the points on the x-axis where the graph cuts the x-axis. In other words, we can say that the zeros of a function are the x-intercepts of its graph. The number of zeros of a polynomial function is equal to the degree of the polynomial.

Zeros of a Function Formula

To find the zeros of a function f(x), we solve the equation f(x) = 0 for x. To find the roots of a function, we can use different methods to factorize the function and then equate it to 0. We can factorize the function using various methods such as:

• Grouping
• Algebraic Identities
• Splitting the Middle Term

For quadratic functions, we can use the discriminant formula to determine if the zeros of the function exist. Then, we can use the r to find the zeros.

Finding Real Zeros of a Function

A real zero of a function f(x) is a real number that satisfies the equation f(x) = 0. In other words, we can say that a real number 'r' is a zero of a function f(x) if f(r) = 0. Let us consider an example below to find the real zeros of a function. Consider f(x) = 4x3 + 16x2 + 7x. Now, we need to find the factors of this function to determine its zeros.

Solution: We have f(x) = 4x3 + 16x2 + 7x

= x(4x2 + 16x + 7)

= x(4x2 + 14x + 2x + 7)

= x[2x(2x + 7) + 1(2x + 7)]

= x (2x + 1) (2x + 7)

Now, to find the real zeros of the function f(x), we have f(x) = 0

x (2x + 1) (2x + 7) = 0

⇒ x = 0 or 2x + 1 = 0 or 2x + 7 = 0

⇒ x = 0 or x = -1/2 or x = -7/2 → All are real numbers

So, the real zeros of the function f(x) = 4x3 + 16x2 + 7x are 0, -1/2 and -7/2.

Zeros of a Function on a Graph

In this section, we will learn to find the zeros of a function using a graph. To find the zeros graphically, we need to determine the x-intercepts of the graph of the function. Let us have a look at the graph below and learn how to find the zeros of a function on a graph.

As we can see in the above image, the graph of the function cuts the x-axis at two points x = -2 and x = 2. So, the zeros of the function y = x2 - 4 are -2 and 2 as the x-intercepts of the function are -2 and 2. Hence, to find the zeros of a function using a graph, we determine its x-intercepts.

Important Notes on Zeros of a Function

• The zeros of a function f(x) are values of x which satisfy the equation f(x) = 0.
• The zeros of a function can be real, rational, complex, or repetitive.
• For the quadratic function, we can find the zeros using the quadratic formula.

Related Articles

• Zero Function
• Zeros of Polynomial
• Polynomial Function