Check Injective / Surjective
🎥 Video Explanation
📝 Question
Let \(A = \{x : -1 \le x \le 1\}\).
\[ f(x)=x|x| \]
- A. a bijection
- B. injective but not surjective
- C. surjective but not injective
- D. neither injective nor surjective
✅ Solution
🔹 Step 1: Case-wise Form
For \(x \ge 0\):
\[ f(x)=x^2 \]
For \(x < 0\):
\[ f(x)=-x^2 \] —
🔹 Step 2: Injective Check
On \([-1,0]\): strictly decreasing
On \([0,1]\): strictly increasing
No two distinct inputs give same output.
✔️ Injective
—🔹 Step 3: Range
\[ [-1,1] \]
Same as codomain.
✔️ Surjective
—🔹 Final Answer
\[ \boxed{\text{Option A: bijection}} \]