Question

Let r be a real number and \(f(x) = \begin{cases}     2x - r & \text{if } x \geq r \\     r & \text{if } x < r \end{cases}\).Then, the equation \(f(x)=f(f(x))\) holds for all real values of x where

• A. \(x≤r\)
• B. \(x≥r\)
• C. \(x>r\)
• D. \(x≠r\)

Answer 1

The Correct Option is A

The correct answer is A: \(x ≤ r\)
Let's analyze the equation \(f(x) = f(f(x))\) for different cases: 
Case 1: \(x < r\) 
In this case, the equation \(f(x) = f(f(x))\) becomes \(f(x) = f(2x - r) \space{since}\space x < r\). Now, from the definition of \(f(x)\), when \(x<r, f(x) = r\). So, we have \(r = f(2x - r)\). 
Case 2: \(x ≥ r\) 
In this case, the equation \(f(x) = f(f(x)) \)becomes\( f(x) = f(x)\) since \(x ≥ r\). This simplifies to \(f(x) = x\), which is true for \(x ≥ r\). 
Now,combining both cases: 
For \(x < r\), we have \(r = f(2x - r)\). 
For \(x ≥ r\), we have \(f(x) = x\). 
Since the equation \(r=f(2x - r)\) holds for \(x < r\) and \(f(x) = x\) holds for \(x ≥ r\), the correct answer is: a. x ≤ r

Answer 2

If, \(x< r\)
\(f(x) =r\)
\(⇒ f(x) = f(f(x))\)
\(⇒r=f(r)\)
\(⇒r= 2r-r\)
\(⇒r=r\)

If,  \(x≥r\)
\(f(x) = 2x-r\)
\(⇒f(x) = f(f(x))\)
\(⇒2x-r = f(2x-r)\)
\(⇒2x-r = 2(2x-r) - r\)
\(⇒2x-r = 4x-3r\)
 \(⇒x=r\)
Hence, \(x≤ r\)

So, the correct option is (A): \(x≤ r\)