Let f(x) =`x^(2)` ,g(x)=tan x and h(x)=ln x
For x = `sqrt(pi)/(2)` what is the value of [ho(gof)(x)?

Options

A. A. 0
B. B. 1
C. C. `(pi)/(4)`
D. D. `(pi)/(2)`

Answer

To find the value of \( h(g(f(x))) \) for \( x = \frac{\sqrt{\pi}}{2} \), we will follow these steps:

### Step 1: Identify the functions
We have the following functions:
- \( f(x) = x^2 \)
- \( g(x) = \tan(x) \)
- \( h(x) = \ln(x) \)

### Step 2: Calculate \( f(x) \)
We need to evaluate \( f(x) \) at \( x = \frac{\sqrt{\pi}}{2} \):
\[ f\left(\frac{\sqrt{\pi}}{2}\right) = \left(\frac{\sqrt{\pi}}{2}\right)^2 = \frac{\pi}{4} \]

### Step 3: Calculate \( g(f(x)) \)
Now we need to find \( g(f(x)) \), which is \( g\left(f\left(\frac{\sqrt{\pi}}{2}\right)\right) = g\left(\frac{\pi}{4}\right) \):
\[ g\left(\frac{\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) = 1 \]

### Step 4: Calculate \( h(g(f(x))) \)
Next, we find \( h(g(f(x))) \), which is \( h(1) \):
\[ h(1) = \ln(1) = 0 \]

### Conclusion
Thus, the value of \( h(g(f(x))) \) at \( x = \frac{\sqrt{\pi}}{2} \) is:
\[ \boxed{0} \]