How to find Domain of a Function

1. Concept Overview

Finding the domain of a function means identifying all values of [x] for which the function is defined and gives real output.

In exams, most mistakes happen because students:

• forget to apply restrictions, or
• apply unnecessary restrictions

So, it is best to follow a fixed method every time.

---

2. Golden Rule for Finding Domain

The domain of a function is the largest set of real numbers for which the given function is defined.

Always:

• start with all real numbers
• then remove only those values that make the function undefined

---

3. Step-by-Step Method (Universal Method)

Whenever you are given a function [f(x)], follow these steps:

Step 1. Assume [x ∈ R]
Step 2. Identify expressions that can cause problems
Step 3. Apply restrictions
Step 4. Write the final domain using interval notation

---

4. Common Cases and How to Handle Them

---

Case 1: Polynomial Functions

Example: [f(x)=x^3−4x+7]

Method:

• Polynomials are defined for all real values

Domain: [R]

---

Case 2: Rational Functions (Fractions)

Example: [f(x)=\dfrac{1}{x−5}]

Method:

1. Denominator must not be zero
2. [x−5≠0 ⇒ x≠5]

Domain: All real numbers except [(x=5)]

---

Case 3: Square Root Functions

Example: [f(x)=\sqrt{3x−2}]

Method:

1. Expression inside square root ≥ 0
2. [3x−2≥0 ⇒ x≥\dfrac{2}{3}]

Domain: [[\dfrac{2}{3},∞)]

---

Case 4: Logarithmic Functions

Example: [f(x)=\log(x−4)]

Method:

1. Argument of logarithm must be positive
2. [x−4>0 ⇒ x>4]

Domain: [(4,∞)]

---

Case 5: Combined Functions

Example: [f(x)=\dfrac{1}{\sqrt{x−1}}]

Method:

1. Square root condition: [x−1>0]
2. Denominator cannot be zero

So: [x−1>0 ⇒ x>1]

Domain: [(1,∞)]

---

5. Domain When Multiple Restrictions Exist

Example: [f(x)=\dfrac{\sqrt{x+2}}{x−1}]

Step-by-Step Solution:

1. Square root condition: [x+2≥0 ⇒ x≥−2]
2. Denominator condition: [x−1≠0 ⇒ x≠1]
3. Combine both conditions

Domain: [[−2,1)∪(1,∞)]

---

6. Domain Expressed Using Intervals

---

7. Common Mistakes to Avoid

• Including values where denominator becomes zero
• Allowing negative values inside square root
• Allowing zero or negative values inside logarithm
• Forgetting to combine multiple conditions

---

8. Conceptual Questions with Solutions

1. Why do we start domain from all real numbers?

Because the domain is the largest possible set unless restricted by the function.

2. Why is denominator never allowed to be zero?

Because division by zero is undefined.

3. Why is square root of negative number not allowed?

Because it does not produce a real value.

4. Why is logarithm of zero undefined?

Because no real power of the base gives zero.

5. Can domain be written in set form instead of intervals?

Yes, but interval notation is preferred in Class 12.

6. What if multiple restrictions contradict each other?

Then the domain may become empty.

7. Is domain always continuous?

No. It may be split into multiple intervals.

8. Can domain include irrational numbers?

Yes, unless restricted by the function.

9. Does simplifying a function change its domain?

Sometimes yes, sometimes no. One must be careful.

10. Why is domain checked before graphing?

Because the graph exists only for valid input values.

---

9. Practice Questions With Step-by-Step Solutions

Question 1.

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

Step-by-Step Solution:

1. Square root condition: [x−3>0]
2. Denominator cannot be zero

Conclusion:
Domain = [(3,∞)]

---

Question 2.

Find the domain of [f(x)=\sqrt{5−x}].

Step-by-Step Solution:

1. Expression inside square root ≥ 0
2. [5−x≥0 ⇒ x≤5]

Conclusion:
Domain = [(−∞,5]]

---

Question 3.

Find the domain of [f(x)=\dfrac{1}{x^2−9}].

Step-by-Step Solution:

1. Denominator ≠ 0
2. [x^2−9=0 ⇒ x=±3]

Conclusion:
Domain = [(−∞,−3)∪(−3,3)∪(3,∞)]

---

Question 4.

Find the domain of [f(x)=\log(2x−1)].

Step-by-Step Solution:

1. Argument of log > 0
2. [2x−1>0 ⇒ x>\dfrac{1}{2}]

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

---

Question 5.

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

Step-by-Step Solution:

1. Square root condition: [x+1≥0 ⇒ x≥−1]
2. Denominator condition: [x+1≠0 ⇒ x≠−1]
3. Combine conditions

Conclusion:
Domain = [(−1,∞)]