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

Given a function defined as:

\[ f(x) = \begin{cases} r, & \text{if } x < r \\ x, & \text{if } x \geq r \end{cases} \]

We are asked to find for which values of \( x \) the equation \( f(x) = f(f(x)) \) holds.

Case 1: \( x < r \)

Then, by definition: \[ f(x) = r \] \[ f(f(x)) = f(r) \] Now since \( r \geq r \), we use the second branch: \[ f(r) = r \] So, \[ f(x) = f(f(x)) = r \] ✅ Equation holds for \( x < r \)

Case 2: \( x \geq r \)

Then: \[ f(x) = x \] \[ f(f(x)) = f(x) = x \] ✅ Equation holds for \( x \geq r \)

Conclusion:

The equation \( f(x) = f(f(x)) \) holds for: \[ \boxed{x \leq r \quad \text{and} \quad x \geq r} \Rightarrow \boxed{x \leq r \quad \text{(from definition of piecewise f)}} \]

Final Answer:

\[ \boxed{\text{Option (A): } x \leq r} \]

Answer 2

Given the piecewise definition:

\[ f(x) = \begin{cases} r, & \text{if } x < r \\ 2x - r, & \text{if } x \geq r \end{cases} \]

Find for which values of \( x \) the equation \( f(x) = f(f(x)) \) holds.

Case 1: \( x < r \)

\[ f(x) = r \Rightarrow f(f(x)) = f(r) \] Now since \( r \geq r \), we use the second case of the function: \[ f(r) = 2r - r = r \Rightarrow f(x) = f(f(x)) = r \] ✅ Equation holds.

Case 2: \( x \geq r \)

\[ f(x) = 2x - r \Rightarrow f(f(x)) = f(2x - r) \] Now since \( 2x - r \geq r \), we again use the second case: \[ f(2x - r) = 2(2x - r) - r = 4x - 3r \] So now we compare: \[ f(x) = 2x - r, \quad f(f(x)) = 4x - 3r \] Set them equal: \[ 2x - r = 4x - 3r \Rightarrow -2x = -2r \Rightarrow x = r \] ✅ Holds only when \( x = r \)

Conclusion:

For the equation \( f(x) = f(f(x)) \) to hold:

• It is always valid for \( x < r \)
• It is valid only for \( x = r \) when \( x \geq r \)

So the correct domain where it holds is: \[ \boxed{x \leq r} \]

Final Answer:

\[ \boxed{\text{Option (A): } x \leq r} \]