The chapter “Relations and Functions” builds the foundation for understanding how mappings behave (injective, surjective, bijective), how functions can be composed, and how algebraic and set-based conditions translate into function properties. These ideas are heavily used in board exams and are also frequent in competitive exams through questions on inverse/involution functions, functional equations, continuity/functional forms, and counting functions with constraints.
25
Minutes
20
Questions
1 / -0
Marking
Q1. Let be defined by where . For which pairs does for all real ?
with any real .
and .
or with .
only.
Q2. For which real values of is the function injective on ?
.
.
.
(all real ).
Q3. Let be defined by and , where . Find the necessary and sufficient condition on for for all real .
.
.
.
.
Q4. Let be continuous and satisfy for all real . Which of the following statements must hold?
(i) is bijective.
(ii) is either strictly increasing or strictly decreasing.
(iii) for all .
Only (i) is true.
(i) and (ii) only.
(i) and (iii) only.
(i), (ii) and (iii) are all true.
Q5. Let satisfy for all real , and suppose for every real . Then which of the following is true?
for all real .
for all real .
is strictly increasing but not the identity.
No such function exists.
Q6. Let be defined by
Find the set .
Q7. Let be linear of the form and satisfy
Which of the following describes all possible linear functions ?
Both and
only
only
No linear function exists
Q8. For a fixed real with , consider the equation
How many distinct real solutions does this equation have?
No real solution
Exactly one real solution
Infinitely many real solutions
Exactly two real solutions
Q9. Let be continuous and satisfy for all real the identity
Then must be of which form?
only
for some constants
for some constant
for some constant
Q10. Let be continuous and satisfy
with . Then
Q11. Let be defined by and by . The domain of the composite function is
Q12. For real parameter , define by . For which real does equal the identity on its domain?
only
or
No real makes involutive
All real
Q13. Let . A function satisfies for all . Let be the number of elements fixed by (i.e., ). Which of the following is the set of all possible values of ?
Q14. Define by if and if . Which of the following statements are true?
(I) is bijective.
(II) is the identity on .
(III) is continuous only at .
Only (I) and (II) are true
Only (I) is true
(I), (II) and (III) are all true
(II) and (III) are true but (I) is false
Q15. How many functions satisfy for all ?
...and 5 more challenging questions available in the interactive simulator.