Relations and Functions Question - CBSE

Question

Let $A = \mathbb{R} \setminus \{3\}$ and $B = \mathbb{R} \setminus \{1\}$. Consider the function $f: A 	o B$ defined by $f(x) = \dfrac{x - 2}{x - 3}$. Show that $f$ is one-one and onto. Hence find $f^{-1}$.

Accepted Answer

One-one proof: $f(x_1)=f(x_2) \Rightarrow x_1=x_2$. [1.5 Marks]
Onto proof: $x = \dfrac{3y-2}{y-1} \in A$ for all $y \in B$. [1.5 Marks]
Bijective $\Rightarrow f$ is invertible. [1.0 Mark]
Inverse $f^{-1}(x) = \dfrac{3x-2}{x-1}$. [1.0 Mark]

---
🎯 Official CBSE Marking Scheme:
Proving one-one property: 1.5 Marks
Proving onto property: 1.5 Marks
Stating bijectivity: 1.0 Mark
Deriving inverse function $f^{-1}(x)$: 1.0 Mark

Question:

Step-by-Step Solution

Master Relations and Functions with Mathbee