Find Set A for Surjection
🎥 Video Explanation
📝 Question
Let \( f:\mathbb{R} \to A \) be defined by
\[ f(x)=\frac{x^2}{x^2+1} \]
Find \(A\) such that \(f\) is surjective.
- A. \(\mathbb{R}\)
- B. \([0,1]\)
- C. \((0,1]\)
- D. \([0,1)\)
✅ Solution
🔹 Step 1: Analyze Function
\[ f(x)=\frac{x^2}{x^2+1} \]
Rewrite:
\[ f(x)=1-\frac{1}{x^2+1} \]
—🔹 Step 2: Minimum Value
At \(x=0\):
\[ f(0)=0 \]
—🔹 Step 3: Maximum Value
As \(x \to \infty\):
\[ f(x) \to 1 \] (but never equals 1)
—🔹 Step 4: Range
\[ [0,1) \]
—🔹 Final Answer
\[ \boxed{\text{Option D}} \]