Check Injective / Surjective
🎥 Video Explanation
📝 Question
Given \( f:[0,\infty) \to \mathbb{R} \),
\[ f(x)=\frac{x}{x+1} \]
- A. one-one and onto
- B. one-one but not onto
- C. onto but not one-one
- D. neither one-one nor onto
✅ Solution
🔹 Step 1: Check Injective
Let \(x_1 \ne x_2\).
\[ f(x)=\frac{x}{x+1} \] is strictly increasing on \([0,\infty)\).
Hence, different inputs give different outputs.
✔️ Function is one-one
—🔹 Step 2: Find Range
\[ f(x)=\frac{x}{x+1}=1-\frac{1}{x+1} \]
As \(x \to 0\): \[ f(0)=0 \]
As \(x \to \infty\): \[ f(x) \to 1 \] (but never equals 1)
Range: \[ [0,1) \]
—🔹 Step 3: Check Onto
Codomain is \(\mathbb{R}\), but range is \([0,1)\).
Many real numbers not covered.
❌ Not onto
—🔹 Final Answer
\[ \boxed{\text{Option B: one-one but not onto}} \]