If `g(x)=(1)/(f(x))andf(x)=x,xne0`, then which one of the following is correct

Options

A. `f(f(f(g(g(f(x))))))=g(g(f(g(f(x)))))`
B. `f(f(g(3(g(f(x))))))=g(g(f(g(f(x))))))`
C. `f(g(f(g(g(f(g(x))))))=g(g(f(g(f(x))))))`
D. `f(f(f(f(f(f(x))))))=f(f(f(g(x)))))`

Explanation

To solve the problem, we need to analyze the functions given and determine which option is correct based on the relationships between \( f(x) \) and \( g(x) \).

### Step-by-step Solution:

1. **Define the Functions**:
We are given:
\[ f(x) = x \quad \text{for } x \neq 0 \]
and
\[ g(x) = \frac{1}{f(x)} \]

2. **Substitute \( f(x) \) into \( g(x) \)**:
Since \( f(x) = x \), we can substitute this into \( g(x) \):
\[ g(x) = \frac{1}{f(x)} = \frac{1}{x} \quad \text{for } x \neq 0 \]

3. **Calculate \( g(f(x)) \)**:
Now, we need to find \( g(f(x)) \):
\[ g(f(x)) = g(x) = \frac{1}{x} \]

4. **Calculate \( f(g(x)) \)**:
Next, we find \( f(g(x)) \):
\[ f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{x} \quad \text{for } x \neq 0 \]

5. **Check the Composition \( f(g(x)) \) and \( g(f(x)) \)**:
We can see that:
\[ g(f(x)) = \frac{1}{x} \quad \text{and} \quad f(g(x)) = \frac{1}{x} \]
Thus, we can conclude:
\[ f(g(x)) = g(f(x)) \]

6. **Evaluate Options**:
We need to evaluate the options given in the question. We will check if the expressions involving \( f \) and \( g \) are equal or not.

- **Option A**: \( f(f(f(g(x)))) = g(f(x)) \)
- **Option B**: \( g(f(g(x))) = f(g(x)) \)
- **Option C**: \( f(g(f(x))) = g(g(x)) \)
- **Option D**: \( g(g(f(x))) = f(f(x)) \)

We will check each option based on our derived results.

- **Option A**:
\[ f(f(f(g(x)))) = f(f(f(\frac{1}{x}))) = f(f(\frac{1}{x})) = f(\frac{1}{x}) = \frac{1}{x} \]
And,
\[ g(f(x)) = \frac{1}{x} \]
So, **Option A is correct**.

- **Option B**:
\[ g(f(g(x))) = g(f(\frac{1}{x})) = g(\frac{1}{x}) = x \]
And,
\[ f(g(x)) = \frac{1}{x} \]
So, **Option B is incorrect**.

- **Option C**:
\[ f(g(f(x))) = f(g(x)) = f(\frac{1}{x}) = \frac{1}{x} \]
And,
\[ g(g(x)) = g(\frac{1}{x}) = x \]
So, **Option C is incorrect**.

- **Option D**:
\[ g(g(f(x))) = g(g(x)) = g(\frac{1}{x}) = x \]
And,
\[ f(f(x)) = f(x) = x \]
So, **Option D is incorrect**.

### Conclusion:
From the analysis, we find that **Option A is correct**.