Relations and Functions Question - CBSE

Question

Consider $f: \mathbb{R}_+ 	o [4, \infty)$ given by $f(x) = x^2 + 4$. Show that $f$ is invertible with the inverse $f^{-1}$ of $f$ given by $f^{-1}(y) = \sqrt{y - 4}$, where $\mathbb{R}_+$ is the set of all non-negative real numbers.

Accepted Answer

One-one: $x_1^2+4=x_2^2+4 \Rightarrow x_1=x_2$. [1.0 Mark]
Onto: $y = x^2+4 \Rightarrow x = \sqrt{y-4} \in \mathbb{R}_+$. [1.0 Mark]
Inverse $f^{-1}(y) = \sqrt{y-4}$. Proved. [1.0 Mark]

---
🎯 Official CBSE Marking Scheme:
Proving one-one property: 1.0 Mark
Proving onto property: 1.0 Mark
Deriving inverse formula: 1.0 Mark

Question:

Step-by-Step Solution

Master Relations and Functions with Mathbee