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

Answer: (A)

$x≤r$

Answer 2

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