What is the Domain of a Function? | Simple Explanation for Class 10

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What is the Domain of a Function (basic introduction)?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The Domain of a function is the set of all possible input values (x-values) for which the function gives a valid, real output. Think of it as the 'allowed ingredients' you can put into a recipe to get a proper dish.

Simple Example

Quick Example

Imagine a mobile recharge shop that only accepts ₹10, ₹50, and ₹100 notes for recharges. Here, the 'allowed ingredients' or input values are ₹10, ₹50, and ₹100. These specific amounts form the domain of the recharge function.

Worked Example

Step-by-Step

Let's find the domain of the function f(x) = 1 / (x - 3).

1. **Identify potential problems:** For a fraction, the denominator cannot be zero. If the denominator is zero, the expression is undefined.
2. **Set the denominator to zero:** Our denominator is (x - 3). So, we set x - 3 = 0.
3. **Solve for x:** Add 3 to both sides: x = 3.
4. **Understand the restriction:** This means that x cannot be 3, because if x is 3, the denominator becomes 3 - 3 = 0, and 1/0 is undefined.
5. **State the domain:** The domain is all real numbers except for 3.

Answer: Domain = All real numbers, x ≠ 3.

Why It Matters

Understanding domain is crucial in fields like AI/ML to know what kind of data models can process, and in Engineering to design systems that operate within safe limits. Doctors use it to determine safe dosages of medicine based on patient data, ensuring inputs are within a valid range.

Common Mistakes

MISTAKE: Assuming all numbers are always allowed as inputs. | CORRECTION: Always check for two main restrictions: denominators cannot be zero, and numbers inside a square root (or any even root) cannot be negative.

MISTAKE: Forgetting to state the domain in terms of 'all real numbers except...' or using interval notation incorrectly. | CORRECTION: Clearly define the set of allowed values. For example, 'x ≠ 5' or 'x ≥ 0'.

MISTAKE: Only considering positive values for x in real-world problems. | CORRECTION: Remember that negative numbers or zero can also be valid inputs unless the context specifically restricts them (e.g., number of students cannot be negative).

Practice Questions

Try It Yourself

QUESTION: Find the domain of the function f(x) = x + 5. | ANSWER: All real numbers

QUESTION: Find the domain of the function g(x) = sqrt(x - 4). | ANSWER: x ≥ 4

QUESTION: A delivery service charges ₹20 for distances up to 5 km, and ₹5 per km for every km beyond 5 km. If 'd' is the distance in km, what is the domain for 'd' if the service only delivers within a 20 km radius? | ANSWER: 0 ≤ d ≤ 20

MCQ

Quick Quiz

Which of the following values is NOT in the domain of the function f(x) = 2 / (x - 7)?

-3

The Correct Answer Is:

The denominator of a fraction cannot be zero. If x = 7, then x - 7 = 0, making the function undefined. All other options result in a valid denominator.

Real World Connection

In the Real World

When you use a food delivery app like Swiggy or Zomato, the app calculates delivery time and cost based on your location and the restaurant's location. The 'domain' here involves valid delivery distances and areas (e.g., within a city or a specific radius), ensuring the system only accepts deliverable orders.

Key Vocabulary

Key Terms

FUNCTION: A rule that assigns each input exactly one output | INPUT: The value given to a function (often 'x') | OUTPUT: The value a function produces for a given input (often 'f(x)' or 'y') | REAL NUMBERS: All numbers on the number line, including positive, negative, zero, fractions, and decimals.

What's Next

What to Learn Next

Now that you understand domain, the next step is to explore the 'Range' of a function. The range tells you all the possible output values a function can produce, building on what inputs are allowed.