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 xcoordinates (xintercepts) where the graph cuts the xaxis. 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 xintercepts 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 xaxis where the graph cuts the xaxis. In other words, we can say that the zeros of a function are the xintercepts 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:
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) = 4x^{3} + 16x^{2} + 7x. Now, we need to find the factors of this function to determine its zeros.
Solution: We have f(x) = 4x^{3} + 16x^{2} + 7x
= x(4x^{2} + 16x + 7)
= x(4x^{2} + 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) = 4x^{3} + 16x^{2} + 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 xintercepts 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 xaxis at two points x = 2 and x = 2. So, the zeros of the function y = x^{2}  4 are 2 and 2 as the xintercepts of the function are 2 and 2. Hence, to find the zeros of a function using a graph, we determine its xintercepts.
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.
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Zeros of a Function Examples

Example 1: Find the zeros of the quadratic function f(x) = x^{2} + 6x + 9.
Solution: To find the zeros of the function f(x) = x^{2} + 6x + 9, we will first find its factors using the algebraic identity (a + b)^{2} = a^{2} + 2ab + b^{2}. We have
f(x) = x^{2} + 6x + 9
= x^{2} + 2 × x × 3 + 3^{2}
= (x + 3)^{2}
Now, f(x) = 0
⇒ (x + 3)^{2} = 0
⇒ (x + 3) = 0 and (x + 3) = 0
⇒ x = 3, 3
Answer: The zeros of f(x) = x^{2} + 6x + 9 are 3 and 3.

Example 2: Find the zeros of the function f(x) = e^{x}  1.
Solution: To find the zeros of f(x) = e^{x}  1, we will equate it to 0. So, we have
f(x) = 0
⇒ e^{x}  1 = 0
⇒ e^{x} = 1
⇒ x = 0  [Because the exponential function is equal to 1 when it is raised to the power of 0]
Answer: Zero of function f(x) = e^{x}  1 is x = 0.

Example 3: Find the zeros of the linear function f(x) = 2x  9.
Solution: To find the zeros of function f(x) = 2x  9, we have
f(x) = 0
⇒ 2x  9 = 0
⇒ 2x = 9
⇒ x = 9/2
Answer: Zero of function f(x) = 2x  9 is x = 9/2
FAQs on Zeros of a Function
What are Zeros of a Function in Math?
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 xaxis where the graph cuts the xaxis.
What are Zeros of a Function on a Graph?
On a graph, the zeros of a function are given by the xintercepts of the graph of the function. The zeros of the function on a graph are the xcoordinates of the points where the graph cuts the xaxis.
What are the 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.
How to Find the Zeros of a Function?
We can find the zeros of a function by equating the function to 0 and then finding the corresponding values of the variable. For polynomials, we can find the zeros by factorizing the polynomial and then equating it to 0 to find the zeros.
How to Find the Zeros of a Quadratic Function?
To find if the real zeros of a quadratic function exist, we use the discriminant formula. To find the zeros of a quadratic function, we use the quadratic formula.
What is the Maximum Number of Zeros of a Quadratic Function?
Since the degree of a quadratic function is two, therefore the maximum number of zeros that a quadratic function can have is 2.
What is the Difference Between the Xintercepts and Zeros of a Function?
The zeros of a function y = f(x) are found where y = 0. So, xintercepts and zeros of a function are the same as these points give the value of the variable of a function where it is equal to 0. An xintercept is a point where the y coordinate is zero and the zero of a function is a value which when substituted into the function gives the value equal to 0.
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