Chapter 1 – Relations and Functions

1. If f(x)=x² on ℝ, determine whether f is one-one. Give a reason.

2. If f:A→B is both one-one and onto, what type of function is f?

3. If f(x)=2x+3, find f⁻¹(9).

4. State whether the relation R={(a,b): a=b} on ℝ is reflexive.

5. If f∘g = I, what can be inferred when both functions are bijections?

6. Find the range of f(x)=|x|, where x∈ℝ.

7. Let f(x)=x/(x+1). Determine the domain of f and explain why x=-1 is excluded.

8. Show that the function f(x)=3x−5 is invertible and obtain its inverse.

9. Let A={1,2,3} and B={2,4,6}. Define a relation by b=2a. Determine whether it represents a function from A to B.

10. If f(x)=(x−2)/(x+3), find f⁻¹(x). Verify your result using f(f⁻¹(x))=x.

11. Let f(x)=x²−4x+7. Determine whether the function is one-one on ℝ. If not, suggest a suitable restricted domain on which it becomes one-one.

12. For f(x)=2x+1 and g(x)=x², compare f∘g and g∘f. Are they equal? Explain.