With f(x) = x + 1 and g(x) = 2x, (f ∘ g)(2) works out to 5, and a function is invertible exactly when it is both one-one and onto, so this Class 12 sheet opens with composition and the inversion condition. The inverse answer (x − 3)/2 follows.
Key Takeaways
(f ∘ g)(2) = 5 when f(x) = x + 1 and g(x) = 2x (Q1).
A function is invertible when it is both one-one and onto (Q2).
Q4 records (x − 3)/2 as an inverse.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.If f(x) = x + 1 and g(x) = 2x, then (f ∘ g)(2) = ?
(a) 4(b) 5(c) 6(d) 3
2.A function f : A → B is invertible if and only if it is:
(a) One-one(b) Onto(c) Both one-one and onto(d) A constant function
3.The function f : R → R given by f(x) = x² is:
(a) One-one on R(b) Onto on R(c) Neither one-one nor onto on R(d) Both one-one and onto on R
4.If f : R → R is given by f(x) = 2x + 3, then f⁻¹(x) = ?
(a) (x + 3)/2(b) (x - 3)/2(c) 2x - 3(d) (3 - x)/2
5.If f(x) = x² and g(x) = √x, then (g ∘ f)(x) for x ≥ 0 is:
(a) x²(b) x(c) √x(d) x⁴
6.The relation R = {(a, a), (b, b), (c, c)} on the set A = {a, b, c} is:
(a) Reflexive only(b) An equivalence relation(c) Transitive only(d) Symmetric only
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Define an equivalence relation on a set.
8.If f(x) = 2x and g(x) = x + 3, find (f ∘ g)(x).
9.State the condition under which a function f : A → B is onto.
10.If f(x) = x³, show that f is one-one on R.
11.Find f⁻¹(x) if f(x) = 3x - 4.
12.If R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, is R transitive? Give a reason.
13.If f : A → B and g : B → C, what is the domain of the composite function g ∘ f?
14.State whether the relation y = x² on the set of integers is symmetric. Justify.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.If f : R → R is given by f(x) = x + 1, find f⁻¹(x) and verify that f(f⁻¹(x)) = x for all x.
16.Let f : R → R be given by f(x) = 5x - 7. Find the value of f⁻¹(13).
17.A function f : R → R is given by f(x) = 2x + 5. Find the number x such that f(x) = 21.
18.If f(x) = x³ and g(x) = 2x + 1, find the value of (f ∘ g)(2).
19.Show that the relation R = {(x, y) : x + y is even} on the set of integers is an equivalence relation, and identify its equivalence classes.
20.If f(x) = 3x + 2, find the value of a such that f(a) = 17.
Answer Key
1. b) 5
2. c) Both one-one and onto
3. c) Neither one-one nor onto on R
4. b) (x - 3)/2
5. b) x
6. b) An equivalence relation
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is (f ∘ g)(2) when f(x) = x + 1 and g(x) = 2x?
5, option (b) in this worksheet (Q1).
When is f : A → B invertible?
When it is both one-one and onto, option (c) in this worksheet (Q2).
What is the inverse of f(x) = 2x + 3?
(x − 3)/2, option (b) in this worksheet (Q4).
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