Introduction to Functions

1. Concept Overview

In Mathematics, especially in Class 12, the idea of a function forms the foundation of almost every chapter—Limits, Continuity, Differentiation, Integration, Differential Equations, and many more.

In simple words:

A function is a rule that assigns exactly one output to each input.

Think of a function as a machine:

• You put something in (input)
• The machine works according to a fixed rule
• You get one definite output

This input–output relationship is what makes Mathematics logical, predictable, and powerful.

---

2. Clear Explanation and Mathematical Definition

(a) Sets Involved in a Function

A function always connects two sets:

• Domain → the set of all possible inputs
• Codomain → the set containing possible outputs

If a function [f] maps elements from set [A] to set [B], we write:

[f : A \to B]

Here:

• [A] is the domain
• [B] is the codomain

If an element [x \in A] is mapped to an element [y \in B], we write:

[y = f(x)]

---

(b) Condition for a Relation to be a Function

A relation [f : A \to B] is called a function if:

Every element of set [A] has exactly one image in set [B].

Important points:

• Every input must have an output
• One input → only one output
• Two different inputs may have the same output (allowed)

---

3. Real-Life Understanding of Functions

Let us understand functions through everyday examples:

1. Student Roll Number → Student Name
   • Each roll number gives only one student name
   • Hence, it is a function
2. Person → Aadhaar Number
   • One person has only one Aadhaar number
   • Function
3. Person → Mobile Numbers
   • One person may have multiple mobile numbers
   • ❌ Not a function

These examples help us understand the “exactly one output” rule clearly.

---

4. Domain, Codomain, and Range

Let [f : A \to B] be a function.

• Domain: Set [A] (all inputs)
• Codomain: Set [B] (possible outputs)
• Range: Set of actual outputs obtained

Mathematically:

[Range \subseteq Codomain]

Example:
If [f(x) = x^2] and domain is all real numbers:

• Domain = [R]
• Codomain = [R]
• Range = [y \ge 0]

---

5. Representation of Functions

A function can be represented in several ways:

1. Algebraic form: [f(x) = 2x + 3]
2. Set form: [(x, y) \text{ such that } y = 2x + 3]
3. Arrow diagram
4. Graphical representation

In higher classes, we mostly use algebraic and graphical forms.

---

6. Important Notes for Class 12 Students

• A function is a special type of relation
• Domain is never empty
• Same output for different inputs is allowed
• Multiple outputs for one input is not allowed
• Understanding functions properly makes calculus very easy

---

7. Important Formulas / Definitions to Remember

---

8. Conceptual Questions with Solutions

1. What makes a relation a function?

A relation becomes a function when every element of the domain is associated with exactly one element of the codomain. This uniqueness condition is essential.

2. Why is the condition “exactly one output” necessary?

This condition ensures predictability and certainty. Without it, a single input could give multiple outputs, which is not acceptable in mathematics.

3. Can two different inputs have the same output in a function?

Yes. A function can be many-to-one. For example, in [f(x)=x^2], both [x=2] and [x=-2] give the same output.

4. Why is one-to-many mapping not allowed in functions?

One-to-many mapping destroys uniqueness of output. Hence, such relations cannot be functions.

5. Is every function a relation?

Yes. A function is a special type of relation with an additional restriction.

6. Can a function exist without a codomain?

No. Every function must map elements from a domain to a codomain.

7. Why can the domain of a function not be empty?

A function represents a mapping. If the domain is empty, no mapping exists, so it cannot be a function.

8. Is it compulsory that every element of the codomain appears in the range?

No. Only those elements that are actually obtained as outputs form the range.

9. Can two different functions have the same domain and range?

Yes. The rule of mapping may be different even if domain and range are the same.

10. Is the function [f(x)=|x|] one-to-one?

No. Since [f(2)=f(-2)=2], it is a many-to-one function.

11. Can a function be constant?

Yes. A constant function assigns the same output to every input.

12. Why is range always a subset of codomain?

Because the codomain contains all possible outputs, while the range contains only actual outputs.

13. Is the relation [(x,y):y^2=x] a function?

No. For a single value of [x], there are two values of [y], so the uniqueness condition fails.

14. Can a function map an element to itself?

Yes. For example, [f(x)=x] maps each element to itself.

15. Why are functions called the foundation of calculus?

Because concepts like limits, continuity, differentiation, and integration are all based on functions.

---

9. FAQ / Common Misconceptions

1. Every relation is a function.

False. Only relations satisfying the exactly one output rule are functions.

2. Domain and range are always the same.

False. The range is usually a subset of the codomain.

3. A function must be one-to-one.

False. Many-to-one functions are also valid.

4. One-to-many relations are also functions.

False. One-to-many relations violate the definition of a function.

5. Every equation represents a function.

False. Only equations that give a unique output for each input represent functions.

6. Codomain and range mean the same thing.

False. Codomain is the set of possible outputs, while range is the set of actual outputs.

7. A function cannot be constant.

False. Constant functions are valid.

8. Graphs are not important while studying functions.

False. Graphs help in visual understanding of functions.

9. Functions are used only in Mathematics.

False. Functions are widely used in Physics, Economics, Computer Science, and Engineering.

10. Changing the domain does not affect the function.

False. The domain plays a crucial role in defining a function.

---

10. Practice Questions

Question 1. State whether the relation [(x, y) : y = 3x + 1] is a function.

Step-by-Step Solution:

1. For each value of [x], there is exactly one value of [y]
2. Hence, the relation satisfies the function condition

Conclusion: It is a function.

---

Question 2. Is the relation [(x, y) : y^2 = x] a function? Explain.

Step-by-Step Solution:

1. For [x = 4], [y = 2] and [y = -2]
2. One input has two outputs

Conclusion: It is not a function.

---

Question 3. Find the domain of the function [f(x) = \dfrac{1}{x – 2}].

Step-by-Step Solution:

1. Denominator cannot be zero
2. [x – 2 \ne 0 \Rightarrow x \ne 2]

Conclusion: Domain is all real numbers except [(x = 2)].

---

Question 4. If [f(x) = x^2], write its range.

Step-by-Step Solution:

1. Square of any real number is non-negative

Conclusion: Range is [y \ge 0].

---

Question 5. Write domain and range of [f(x) = |x|].

Step-by-Step Solution:

1. Absolute value is defined for all real numbers
2. Output is always non-negative

Conclusion:

• Domain = [R]
• Range = [y \ge 0]

---

Final Remark

Understanding functions deeply is the key to mastering Class 12 Mathematics. Once this concept is clear, calculus becomes logical and enjoyable.