If `f(x)=sqrt(|3^(x)-3^(1)|-2) and `g(x)=tan pi x,` then domain of fog(x) , is

Options

A. `[n+(1)/(3),n+(1)/(2)] cup [n+(1)/(2),n+1], n in Z `
B. `(nx+(1)/(4),n+(1)/(2)) cup (n+(1)/(2),n+1), n cup Z `
C. `(n+(1)/(4), n+(1)/(2)) cup [n-(1)/(2),n+1],n in Z `
D. `[n+(1)/(4),x +(1)/(2)) cup (n+(1)/(2), n+2), n in Z `

Answer

To find the domain of the composite function \( f(g(x)) \), where \( f(x) = \sqrt{|3^x - 3^1| - 2} \) and \( g(x) = \tan(\pi x) \), we will follow these steps:

### Step 1: Determine the domain of \( g(x) \)
The function \( g(x) = \tan(\pi x) \) is defined for all \( x \) except where \( \cos(\pi x) = 0 \). This occurs at:
\[ \pi x = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]
Thus, the values of \( x \) where \( g(x) \) is undefined are:
\[ x = \frac{1}{2} + n \quad \text{for } n \in \mathbb{Z} \]

### Step 2: Find the expression for \( f(g(x)) \)
Now we substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f(\tan(\pi x)) = \sqrt{|3^{\tan(\pi x)} - 3^1| - 2} \]
This requires that the expression inside the square root is non-negative:
\[ |3^{\tan(\pi x)} - 3| - 2 \geq 0 \]
This simplifies to:
\[ |3^{\tan(\pi x)} - 3| \geq 2 \]

### Step 3: Solve the inequality
This absolute value inequality can be split into two cases:
1. \( 3^{\tan(\pi x)} - 3 \geq 2 \)
2. \( 3 - 3^{\tan(\pi x)} \geq 2 \)

**Case 1:**
\[ 3^{\tan(\pi x)} - 3 \geq 2 \implies 3^{\tan(\pi x)} \geq 5 \implies \tan(\pi x) \geq \log_3(5) \]

**Case 2:**
\[ 3 - 3^{\tan(\pi x)} \geq 2 \implies 3^{\tan(\pi x)} \leq 1 \implies \tan(\pi x) \leq 0 \]

### Step 4: Combine the results
Now we need to find the intervals of \( x \) that satisfy both conditions:
1. \( \tan(\pi x) \geq \log_3(5) \)
2. \( \tan(\pi x) \leq 0 \)

The function \( \tan(\pi x) \) is periodic with period 1. The intervals where \( \tan(\pi x) \leq 0 \) are:
\[ x \in [n, n + \frac{1}{2}) \quad \text{for } n \in \mathbb{Z} \]

### Step 5: Exclude points where \( g(x) \) is undefined
We also need to exclude points where \( g(x) \) is undefined:
\[ x \neq \frac{1}{2} + n \quad \text{for } n \in \mathbb{Z} \]

### Final Domain
The final domain of \( f(g(x)) \) is the intersection of the intervals derived from the inequalities and the exclusion of the points where \( g(x) \) is undefined.