Question 7:- Let f : W → W, be defined as f(x) = x – 1, if x is odd and f(x) = x + 1, Show that f is bijective.
Solution: One-one :-
Case I : When x, y are even number
Now, f(x) = f(y) ⇒ x + 1 = y + 1
⇒ x = y
i.e. f is one-one.
Case II: When x, y are odd number
Now, f(x) = f(y) ⇒ x – 1 = y – 1
⇒ x = y
i.e. f is one-one
Case III: When x is odd and y is even number
Then, x ≠ y. Also,in this case f(x) is even and f(y) is odd and so
f(x) ≠ f(y)
i.e. x ≠ y ⇒ f(x) ≠ f(y)
i.e., f is one-one.
Case IV: When x is even and y is odd number
Then, x ≠ y. Also,in this case f(x) is odd and f(y) is even and so
f(x) ≠ f(y)
i.e. x ≠ y ⇒ f(x) ≠ f(y)
i.e., f is one-one.
Onto:
Given, $f(x) = \begin{cases} x – 1, & \text{if x is odd} \\ x + 1, &\text{if x is even} \end{cases}$
⇒ For every even number ‘y’ of codomain ∃ odd number y + 1 in domain and for every odd number y of codomain there exists even number y – 1 in domain i.e., f is onto function.
Hence, f is one-one onto i.e., invertible function.
Other question:-
Question 1:- Show that the relation R on the set R of real numbers, defined as R= {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive.
Solution: See full solution
Question 2: Check whether the relation R in R defined as R={(a, b): a ≤ b³} is reflexive, symmetric or transitive.
Solution: See full solution
Question 3:- Consider $\large f:R_+ \rightarrow [-9, \infty)$ given by $\large f(x) = 5x^2+6x-9$ prove that f is bijective.
Solution: See full solution
Question 4:- Consider $f:R_+$ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible. [CBSE(AI) 2013]
Solution: See full solution
Question 5:- Let f : R – {-4/3} → R be a function defined as f(x) = 4x/(3x+4). Show that, f:R – {-4/3} → Range of f, f is one-one and onto. [CBSE 2017(C)]
Solution:- See full solution
Question 6:- Let A = R – {3}, B = R – {1}. If f : A → B be defined by $f(x) = \dfrac{x – 2}{x – 3}$, ∀x ∈ A. Then, show that f is bijective.
Solution:- See full solution