What is the Zeroes of a Function?

S3-SA5-0105

Grade Level:

Class 10

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The 'zeroes' of a function are the input values (usually 'x' values) for which the function's output (usually 'y' or 'f(x)' value) becomes zero. Think of them as the points where the graph of the function crosses or touches the X-axis.

Simple Example

Quick Example

Imagine you have a function that tells you how much money you have left after buying 'x' number of samosas, where each samosa costs Rs 10 and you started with Rs 50. The function would be f(x) = 50 - 10x. The 'zero' of this function would be when you have Rs 0 left. This happens when x = 5, meaning you bought 5 samosas.

Worked Example

Step-by-Step

Let's find the zeroes of the function f(x) = 2x - 6.

1. To find the zeroes, we set the function's output to zero: f(x) = 0.
---2. So, we write the equation: 2x - 6 = 0.
---3. Now, we solve for x. Add 6 to both sides: 2x = 6.
---4. Divide both sides by 2: x = 6 / 2.
---5. Therefore, x = 3.
---The zero of the function f(x) = 2x - 6 is x = 3.

Why It Matters

Understanding zeroes helps scientists and engineers predict outcomes, like when a rocket will hit the ground or when a company's profit will be zero. In AI/ML, it helps algorithms find optimal solutions. Many careers in data science, physics, and economics rely on finding these critical points.

Common Mistakes

MISTAKE: Confusing zeroes with the y-intercept. Zeroes are when y=0 (x-intercepts) | CORRECTION: Remember that 'zeroes' means the function's value is zero, so you set f(x) = 0, not x = 0.

MISTAKE: Not solving the equation completely, or making calculation errors. For example, in 2x - 6 = 0, solving as 2x = -6. | CORRECTION: Always check your basic arithmetic (addition, subtraction, multiplication, division) when isolating 'x'.

MISTAKE: Thinking all functions have only one zero. | CORRECTION: Some functions, especially quadratic ones (like x^2 - 4), can have multiple zeroes (e.g., x = 2 and x = -2).

Practice Questions

Try It Yourself

QUESTION: Find the zero of the function g(x) = 3x + 9. | ANSWER: x = -3

QUESTION: What are the zeroes of the function h(x) = x^2 - 9? (Hint: There are two!) | ANSWER: x = 3 and x = -3

QUESTION: A mobile data pack costs Rs 100 and gives you 2 GB. If you use 0.5 GB per day, the data remaining after 'd' days is R(d) = 2 - 0.5d. After how many days will your data remaining be zero? | ANSWER: 4 days

MCQ

Quick Quiz

For the function f(x) = 5x - 15, what is its zero?

x = 15

x = 3

x = -3

x = 0

The Correct Answer Is:

To find the zero, set f(x) = 0. So, 5x - 15 = 0. Adding 15 to both sides gives 5x = 15. Dividing by 5 gives x = 3. Therefore, x = 3 is the zero.

Real World Connection

In the Real World

Imagine a startup company calculating its 'break-even point'. This is exactly finding the zero of a profit function (Profit = Revenue - Cost). When profit is zero, the company is neither making nor losing money. Businesses use this to set prices and plan production, similar to how local kirana stores manage their stock.

Key Vocabulary

Key Terms

FUNCTION: A rule that assigns each input exactly one output | X-INTERCEPT: The point where a graph crosses the X-axis | EQUATION: A statement that two expressions are equal | VARIABLE: A symbol (like x) that represents a quantity that can change | ROOTS: Another name for the zeroes of a polynomial function

What's Next

What to Learn Next

Now that you understand zeroes, you're ready to explore how to find zeroes for more complex functions, like quadratic equations using factorization or the quadratic formula. This will open doors to solving many real-world problems!