Class 12 Maths Chapter 1 Relations and Functions – Exercise 1.1 NCERT Solutions

Introduction

Exercise 1.1 revises the basic concepts of relations and functions. Students learn how to identify relations, functions, domains, ranges, and types of mappings. This exercise builds the foundation for advanced topics like composition, inverse functions, and binary operations.

Formulas Used

1. Relation: A subset of $A\times B$.

2. Function: A relation where each element of domain has exactly one image in codomain.

3. Domain: Set of all input values.

4. Range: Set of all output values.

5. Types of Functions: One‑one, onto, bijective.

Students Frequently Make Mistakes

• Confusing relation with function.

• Forgetting that function must assign unique output for each input.

• Errors in identifying domain restrictions (like square root or denominator).

• Mixing up range with codomain.

• Not checking one‑one and onto conditions properly.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. If $A=\{1,2,3\},B=\{a,b\}$, list all relations from $A$ to $B$. Total relations = subsets of $A\times B$. $\mathrm{\mid }A\times B\mathrm{\mid }=6⟹2^6=64$ relations. ✔

Q2. If $A=\{1,2\},B=\{x,y\}$, list all functions from $A$ to $B$. Each of 2 elements in $A$ can map to 2 elements in $B$. Total functions = $2^2=4$. Functions: $\{(1,x),(2,x)\},\{(1,y),(2,y)\},\{(1,x),(2,y)\},\{(1,y),(2,x)\}$.

Q3. Find domain and range of $f(x)=x^2$. Domain: $\mathbb{R}$. Range: $[0,\mathrm{\infty })$.

Q4. Find domain and range of $f(x)=\sqrt{x}$. Domain: $[0,\mathrm{\infty })$. Range: $[0,\mathrm{\infty })$.

Q5. Find domain and range of $f(x)=\frac{1}{x}$. Domain: $\mathbb{R}\setminus \{0\}$. Range: $\mathbb{R}\setminus \{0\}$.

Q6. Find domain and range of $f(x)=\frac{1}{x^2+1}$. Domain: $\mathbb{R}$. Range: $(0,1]$.

Q7. Check if relation $R=\{(1,2),(2,3),(3,3)\}$ from $A=\{1,2,3\}$ to $B=\{2,3\}$ is function. Each element of $A$ has unique image. ✔ Hence, function.

Q8. Check if relation $R=\{(1,2),(1,3)\}$ is function. Element 1 maps to both 2 and 3. ✘ Not a function.

Q9. If $f(x)=2x+3$, find domain and range. Domain: $\mathbb{R}$. Range: $\mathbb{R}$.

Q10. If $f(x)=\mathrm{\mid }x\mathrm{\mid }$, find domain and range. Domain: $\mathbb{R}$. Range: $[0,\mathrm{\infty })$.

FAQs (10)

FAQ1. What is relation? Subset of $A\times B$.

FAQ2. What is function? Relation with unique output for each input.

FAQ3. How to find domain? Set of all valid inputs.

FAQ4. How to find range? Set of all possible outputs.

FAQ5. What is codomain? Target set of function.

FAQ6. Difference between range and codomain? Range ⊆ Codomain.

FAQ7. What is one‑one function? Distinct inputs → distinct outputs.

FAQ8. What is onto function? Range = codomain.

FAQ9. What is bijective function? Function that is both one‑one and onto.

FAQ10. Why is Exercise 1.1 important? It builds foundation for advanced functions and calculus.

Conclusion

Exercise 1.1 has 10 solved questions and 10 FAQs that strengthen your understanding of relations, functions, domain, and range. This builds the foundation for advanced topics in Class 12 Maths.

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