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Kumar Rohan

Physics and Mathematics

Domain of a Function

1. Concept Overview

The domain of a function tells us which input values are allowed for a function.

In simple words:

The domain is the set of all values of [x] for which the function is defined and meaningful.

Before evaluating, graphing, or differentiating any function, the first step is always to find its domain.

2. Meaning of Domain

If a function is written as [f(x)], then:

[x] is the input
[f(x)] is the output

The domain includes all real values of [x] for which [f(x)] exists.

Mathematically:
If [f : A → B], then
Domain = A

3. Why Domain is Important

Prevents undefined expressions
Ensures real values only
Essential for graphs, limits, continuity, and differentiation
Different domains can create different functions even with the same formula

4. General Rules to Find Domain

Unless stated otherwise, the domain is taken as the largest possible set of real numbers for which the function is defined.

Key Restrictions

Denominator cannot be zero
Expression inside square root must be non-negative
Logarithmic argument must be positive

5. Domain of Common Functions

(a) Polynomial Functions

Example: [f(x)=x^2−3x+1]

Polynomials are defined for all real values

Domain: [R]

(b) Rational Functions

Example: [f(x)=\dfrac{1}{x−2}]

Step:

Denominator ≠ 0
[x−2≠0 ⇒ x≠2]

Domain: All real numbers except [(x=2)]

(c) Square Root Functions

Example: [f(x)=\sqrt{x−3}]

Step:

Expression inside root ≥ 0
[x−3≥0 ⇒ x≥3]

Domain: [[3,∞)]

(d) Logarithmic Functions

Example: [f(x)=\log(x−1)]

Step:

Argument of log > 0
[x−1>0 ⇒ x>1]

Domain: [(1,∞)]

6. Domain Expressed Using Intervals

Domains are often written using interval notation:

[x>2] → [(2,∞)]
[x≤5] → [(−∞,5]]
[x≠3] → [(−∞,3)∪(3,∞)]

7. Conceptual Questions with Solutions

1. What is meant by the domain of a function?

The domain is the set of all input values for which the function is defined.

2. Why can division by zero not be allowed in domain?

Division by zero is undefined, so such values must be excluded.

3. Why must the expression inside a square root be non-negative?

Because the square root of a negative number is not a real number.

4. Why must the argument of a logarithm be positive?

Logarithms of zero or negative numbers are undefined.

5. Can the domain of a function be an open interval?

Yes. Many functions are defined only on open intervals.

6. Can the same formula represent different functions?

Yes. Changing the domain changes the function.

7. Is the domain always mentioned explicitly?

No. If not mentioned, we take the largest possible real domain.

8. Is domain related to graph of a function?

Yes. The graph exists only for values in the domain.

9. Can the domain be a single number?

Yes. A function can be defined for only one input value.

10. Why is domain studied before range?

Because range depends on domain.

11. Is [f(x)=x^2] defined for negative values?

Yes. Squaring a negative number is defined.

12. Can irrational numbers belong to domain?

Yes, unless restricted by the function.

13. Why is domain important in calculus?

Because limits, continuity, and derivatives depend on valid input values.

14. Can a domain be empty?

No. A function must have at least one input.

15. Is domain always a subset of real numbers?

In Class 12 Mathematics, yes.

8. FAQ / Common Misconceptions

1. Domain means output values.

False. Domain refers to input values.

2. Domain of every function is [R].

False. Many functions have restricted domains.

3. Negative numbers are never allowed in domain.

False. They are allowed unless restricted.

4. Domain and range are the same.

False. They represent different sets.

5. Logarithmic functions accept zero.

False. Logarithm of zero is undefined.

6. Square root always allows all values of x.

False. The expression inside must be non-negative.

7. If domain is not written, it is empty.

False. We assume the largest possible domain.

8. Domain does not affect the graph.

False. Graph exists only over the domain.

9. Rational functions have no restrictions.

False. Denominator cannot be zero.

10. Domain is not important for exams.

False. Domain-based questions are very common.

9. Practice Questions: Domain of a Function

Question 1.

Find the domain of [f(x)=x^2+5x+1].

Step-by-Step Solution:

Polynomial functions are defined for all real values.

Conclusion:
Domain = [R]

Question 2.

Find the domain of [f(x)=\dfrac{1}{x−3}].

Step-by-Step Solution:

Denominator must not be zero.
[x−3≠0 ⇒ x≠3]

Conclusion:
Domain = All real numbers except [(x=3)]

Question 3.

Find the domain of [f(x)=\sqrt{2x−1}].

Step-by-Step Solution:

Expression inside square root ≥ 0
[2x−1≥0 ⇒ x≥\dfrac{1}{2}]

Conclusion:
Domain = [[\dfrac{1}{2},∞)]

Question 4.

Find the domain of [f(x)=\log(x+2)].

Step-by-Step Solution:

Argument of log must be positive
[x+2>0 ⇒ x>−2]

Conclusion:
Domain = [(−2,∞)]

Question 5.

Find the domain of [f(x)=\dfrac{1}{x^2−4}].

Step-by-Step Solution:

Denominator ≠ 0
[x^2−4=0 ⇒ x=±2]

Conclusion:
Domain = [(−∞,−2)∪(−2,2)∪(2,∞)]