If `f(x)=(25-x^(4))^(1//4)"for "0 lt x lt sqrt5, "then"f(f((1)/(2)))=`

Options

A. `2^(-4)`
B. `2^(-3)`
C. `2^(-2)`
D. `2^(-1)`

Answer

To solve the problem, we need to find \( f(f(\frac{1}{2})) \) where \( f(x) = (25 - x^4)^{\frac{1}{4}} \) for \( 0 < x < \sqrt{5} \).

### Step-by-Step Solution:

1. **Calculate \( f(\frac{1}{2}) \)**:
\[ f\left(\frac{1}{2}\right) = \left(25 - \left(\frac{1}{2}\right)^4\right)^{\frac{1}{4}} \]
First, compute \( \left(\frac{1}{2}\right)^4 \):
\[ \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]
Now substitute this back into the function:
\[ f\left(\frac{1}{2}\right) = \left(25 - \frac{1}{16}\right)^{\frac{1}{4}} = \left(\frac{400}{16} - \frac{1}{16}\right)^{\frac{1}{4}} = \left(\frac{399}{16}\right)^{\frac{1}{4}} \]
This simplifies to:
\[ f\left(\frac{1}{2}\right) = \frac{(399)^{\frac{1}{4}}}{2} \]

2. **Calculate \( f(f(\frac{1}{2})) \)**:
Now we need to find \( f\left(f\left(\frac{1}{2}\right)\right) \):
\[ f\left(f\left(\frac{1}{2}\right)\right) = f\left(\frac{(399)^{\frac{1}{4}}}{2}\right) \]
Substitute \( x = \frac{(399)^{\frac{1}{4}}}{2} \) into the function:
\[ f\left(\frac{(399)^{\frac{1}{4}}}{2}\right) = \left(25 - \left(\frac{(399)^{\frac{1}{4}}}{2}\right)^4\right)^{\frac{1}{4}} \]
Calculate \( \left(\frac{(399)^{\frac{1}{4}}}{2}\right)^4 \):
\[ \left(\frac{(399)^{\frac{1}{4}}}{2}\right)^4 = \frac{(399)}{16} \]
Now substitute this back:
\[ f\left(\frac{(399)^{\frac{1}{4}}}{2}\right) = \left(25 - \frac{399}{16}\right)^{\frac{1}{4}} = \left(\frac{400}{16} - \frac{399}{16}\right)^{\frac{1}{4}} = \left(\frac{1}{16}\right)^{\frac{1}{4}} \]
This simplifies to:
\[ f\left(f\left(\frac{1}{2}\right)\right) = \frac{1}{2} \]

### Final Answer:
Thus, \( f(f(\frac{1}{2})) = \frac{1}{2} \).