Let f : X → Y be an invertible function. Show that the inverse of f-1 is f, i.e., (f-1)-1 = f.
Let f : X → Y be an invertible function. Show that the inverse of f-1 is f, i.e., (f-1)-1 = f.
Solution and Explanation
Let f : X → Y be an invertible function.
Then, there exists a function g: Y → X such that gof = IX and fog = IY
Here, f-1 = g.
Now, gof = IX and fog = IY
⇒ f-1 of = IX and fof -1= IY
Hence, f-1 : Y → X is invertible and f is the inverse of f-1
i.e., (f-1)-1 = f.
Top Questions on Relations and Functions
- Let \( R \) be a relation on the set of real numbers \( \mathbb{R} \) defined as \[ R = \{(x, y) : x - y + \sqrt{3} \text{ is an irrational number}, x, y \in \mathbb{R} \}. \] Verify \( R \) for reflexivity, symmetry, and transitivity.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and Functions
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and Functions
- The function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = e^x + e^{-x} \) is:
- WBJEE - 2024
- Mathematics
- Relations and Functions
- For every real number \(x \neq -1\), let \(f(x) = \frac{x}{x+1}\). Write \(f_1(x) = f(x)\) and for \(n \geq 2\), \(f_n(x) = f(f_{n-1}(x))\). Then \(f_1(-2), f_2(-2), \ldots, f_n(-2)\) must be:
- WBJEE - 2024
- Mathematics
- Relations and Functions
- Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 10 - x^2 \), then:
- CUET (UG) - 2024
- Mathematics
- Relations and Functions
Questions Asked in CBSE CLASS XII exam
- Arun, Bashir and Joseph were partners in a firm sharing profits and losses in the ratio of 5 : 3 : 2. They admitted Daksh as a new partner who acquired his share entirely from Arun. If Arun sacrificed \( \frac{1}{5} \)th from his share to Daksh, Daksh's share in the profits of the firm will be :
- CBSE CLASS XII - 2025
- Partnership Accounts
- Assertion (A): Cu cannot liberate \( H_2 \) on reaction with dilute mineral acids. Reason (R): Cu has positive electrode potential.
- CBSE CLASS XII - 2025
- Electrochemistry
Derry in On the Face of it is a victim of self-pity. Analyse Derry’s behaviour in the light of the above statement.
- CBSE CLASS XII - 2025
- Literature
"___ how little changes in the environment can have big repercussions" Tishani Doshi in Journey to the End of the Earth gives an awakening call for man. Analyse the theme of the lesson in the light of the above statement.
- CBSE CLASS XII - 2025
- Literature
- Select the correct option from those given in brackets, to fill in the blank: The incident made the speaker realize that the discrimination was ____ (dehumanizing / elevating).
- CBSE CLASS XII - 2025
- Literature