If `f(x) = 1//x, g(x) = sqrt(x)` for all `x in (0,oo)`, then find (gof)(x).

Answer

To find the composition of the functions \( g(f(x)) \), we will follow these steps:

### Step 1: Identify the functions
We are given two functions:
- \( f(x) = \frac{1}{x} \)
- \( g(x) = \sqrt{x} \)

### Step 2: Write the composition
The composition \( g(f(x)) \) means we will substitute \( f(x) \) into \( g(x) \). This can be expressed as:
\[ g(f(x)) = g\left(\frac{1}{x}\right) \]

### Step 3: Substitute \( f(x) \) into \( g(x) \)
Now, we substitute \( f(x) \) into \( g(x) \):
\[ g\left(\frac{1}{x}\right) = \sqrt{\frac{1}{x}} \]

### Step 4: Simplify the expression
We can simplify \( \sqrt{\frac{1}{x}} \):
\[ \sqrt{\frac{1}{x}} = \frac{1}{\sqrt{x}} \]

### Step 5: Write the final result
Thus, the composition \( g(f(x)) \) is:
\[ g(f(x)) = \frac{1}{\sqrt{x}} \]

### Final Answer:
\[ g(f(x)) = \frac{1}{\sqrt{x}} \]

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