Find Set B for Bijectivity
🎥 Video Explanation
📝 Question
Let \( f:[2,\infty) \to B \) be defined by
\[ f(x)=x^2-4x+5 \]
Find \(B\) such that \(f\) is bijective.
- A. \(\mathbb{R}\)
- B. \([1,\infty)\)
- C. \([4,\infty)\)
- D. \([5,\infty)\)
✅ Solution
🔹 Step 1: Convert to Vertex Form
\[ f(x)=x^2-4x+5 \]
\[ = (x-2)^2 + 1 \]
—🔹 Step 2: Domain Analysis
Given domain: \([2,\infty)\)
Function is increasing on this interval.
⇒ One-one
—🔹 Step 3: Range
Minimum at \(x=2\):
\[ f(2)=1 \]
As \(x \to \infty\), \(f(x) \to \infty\)
Range: \[ [1,\infty) \]
—🔹 Final Answer
\[ \boxed{\text{Option B}} \]