Find Set \(X\) for Invertibility
🎥 Video Explanation
📝 Question
Let \( f:[2,\infty)\to X \),
\[ f(x)=4x-x^2 \]
- (a) \([2,\infty)\)
- (b) \((-\infty,2]\)
- (c) \((-\infty,4]\)
- (d) \([4,\infty)\)
✅ Solution
🔹 Step 1: Rewrite
\[ f(x)=4x-x^2 \]
\[ =-(x^2-4x) =-(x-2)^2+4 \] —
🔹 Step 2: Domain Analysis
Given domain: \([2,\infty)\)
Function is decreasing on this interval ⇒ one-one
—🔹 Step 3: Range
Maximum at \(x=2\):
\[ f(2)=4 \]
As \(x\to\infty\), \(f(x)\to -\infty\)
Range: \[ (-\infty,4] \]
—🔹 Final Answer
\[ \boxed{\text{Option (c)}} \]