Check Injective / Surjective / Bijective
🎥 Video Explanation
📝 Question
Given \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x) = x + \sqrt{x^2} \] Find whether the function is:
- A. Injective
- B. Surjective
- C. Bijective
- D. None of these
✅ Solution
🔹 Step 1: Simplify
\[ \sqrt{x^2} = |x| \]
So, \[ f(x) = x + |x| \]
—🔹 Step 2: Case-wise
Case 1: \(x \ge 0\)
\[ f(x) = x + x = 2x \]
Case 2: \(x < 0\)
\[ f(x) = x – x = 0 \]
—🔹 Step 3: Check Injective
For all negative \(x\), \(f(x)=0\).
Different inputs → same output ⇒ ❌ Not injective
—🔹 Step 4: Check Surjective
Range is: \[ [0,\infty) \]
But codomain is \(\mathbb{R}\).
Negative values are not covered ⇒ ❌ Not surjective
—🔹 Final Answer
\[ \boxed{\text{Option D: None of these}} \]