Zeros of a Function - Definition, Formula, Graph, Examples

Understanding the zeros of a function is fundamental in mathematics, especially in algebra, calculus, and applied sciences. This concept is not only essential for solving equations but also serves as a building block for graphing functions and analyzing their behavior.

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Definition

The values of the variable x for which the function f(x) = 0 are known as the zeros of a function. Put more simply, these are the locations where the function's graph crosses the x-axis. In terms of mathematics,
𝑓(𝑥)=0
The zeros are sometimes referred to as the equation's roots or solutions.

Formula

We solve the following equation to determine the zeros:
𝑓(𝑥) = 0

The process of locating zeros varies depending on the type of function:

Linear functions (f(x) = ax + b): The formula for the zero is 𝑥 = −𝑏/𝑎.
Quadratic functions (f(x) = +bx+c): The quadratic formula can be used to find the zeros.
Polynomial functions: Factoring, synthetic division, or numerical techniques can be applied.
Trigonometric functions: Find solutions to equations such as cos(𝑥) = 0 or sin(𝑥) = 0.

Graphical Representation

The x-coordinates of the places on a graph where a curve contacts or crosses the x-axis are the zeros of that function.

The graph of a linear function is typically a straight line with one zero.
A parabola representing a quadratic function can contain zeros of 0, 1, or 2, depending on whether it touches or crosses the x-axis.
The behavior of the graph at the zeros of higher-degree polynomials is dependent on their multiplicity (even or odd).

Applications

Graphing Functions: By showing the point at which a function meets the x-axis, zeros aid in the precise drawing of graphs.
Engineering and physics: Locating zeros is crucial for resolving equilibrium issues and streamlining procedures.
Economics: Breakeven points are shown by zeros in the profit or cost functions.

The foundation for comprehending a function's behavior and applications is its zeros. The idea offers useful insights in a variety of domains, whether it is applied to graph interpretation or equation solving. Gaining a thorough understanding of how to discover zeros improves one's ability to solve problems and be proficient in mathematics.

FAQs 

Q1: What are the zeros of a function?

Ans: The values of the variable x for which the function f(x) = 0 are known as the zeros of a function.

Q2: How many zeros can a function have?

Ans: The degree of a function is equal to the greatest number of zeros it can contain:

• Solving equations in algebra and calculus.
• Determining critical points for graphing functions.
• Analyzing real-world problems, such as finding breakeven points in economics or equilibrium states in physics.

Q4: How can I locate a function's zeros using a graph?
Ans: Find the locations where the graph and the x-axis cross. These points' zeros are their x-coordinates. The associated zero has an even multiplicity if the graph contacts the x-axis without crossing it.

Q5: What is the formula for zeros of a function?
Ans: We solve the following equation to determine the zeros:
𝑓(𝑥) = 0
The process of locating zeros varies depending on the type of function:

• Linear functions (f(x) = ax + b): The formula for the zero is 𝑥 = −𝑏/𝑎.
• Quadratic functions (f(x) = +bx+c): The quadratic formula can be used to find the zeros.
• Polynomial functions: Factoring, synthetic division, or numerical techniques can be applied.
• Trigonometric functions: Find solutions to equations such as cos(𝑥) = 0 or sin(𝑥) = 0.

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