Question

Let \( f:(-1,1)\to(-1,1) \) be continuous, \( f(x)=f(x^2) \), and \( f(0)=\frac{1}{2} \). Find \( 4f\left(\frac{1}{4}\right) \)

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)
• E. \( 5 \)

Hint:

Repeated squaring inside (-1,1) converges to 0.

Answer 1

The Correct Option is B

Concept: Repeated squaring leads to 0.

Step 1: Use: \[ f(x)=f(x^2) \]

Step 2: Apply repeatedly: \[ f\left(\frac{1}{4}\right)=f\left(\frac{1}{16}\right)=f\left(\frac{1}{256}\right) \]

Step 3: Continue: \[ \to f(0) \]

Step 4: So: \[ f\left(\frac{1}{4}\right)=\frac{1}{2} \]

Step 5: Compute: \[ 4 \times \frac{1}{2} = 2 \]