Check Injective / Surjective
🎥 Video Explanation
📝 Question
Let \( A = B = \{x \in \mathbb{R} : -1 \le x \le 1\} \).
Function: \[ f(x) = x|x| \]
- A. injective but not surjective
- B. surjective but not injective
- C. bijective
- D. none of these
✅ Solution
🔹 Step 1: Case-wise Form
Case 1: \(x \ge 0\)
\[ f(x)=x^2 \]
Case 2: \(x < 0\)
\[ f(x)=-x^2 \] —
🔹 Step 2: Check Injective
On \([-1,0]\): strictly decreasing
On \([0,1]\): strictly increasing
Across whole domain, no two different inputs give same output.
✔️ Function is injective
—🔹 Step 3: Check Surjective
Range: \[ [-1,1] \]
Same as codomain \(B\)
✔️ Function is surjective
—🔹 Final Answer
\[ \boxed{\text{Option C: bijective}} \]