Consider a function `f(x)=(3x+4)/(x-2), xne2`. Find a function g(x) on a suitable domain such that :
(gof) (x) = x = (fog) (x).

Answer

To find a function \( g(x) \) such that \( (g \circ f)(x) = x \) and \( (f \circ g)(x) = x \), we start with the function given:

\[ f(x) = \frac{3x + 4}{x - 2}, \quad x \neq 2 \]

### Step 1: Find the Inverse of \( f(x) \)

To find \( g(x) \), we need to find the inverse of \( f(x) \). We start by setting \( y = f(x) \):

\[ y = \frac{3x + 4}{x - 2} \]

Now, we will solve for \( x \) in terms of \( y \).

### Step 2: Rearranging the Equation

Multiply both sides by \( x - 2 \):

\[ y(x - 2) = 3x + 4 \]

Expanding this gives:

\[ yx - 2y = 3x + 4 \]

### Step 3: Collecting Like Terms

Rearranging the equation to isolate \( x \):

\[ yx - 3x = 2y + 4 \]

Factoring out \( x \):

\[ x(y - 3) = 2y + 4 \]

### Step 4: Solving for \( x \)

Now, divide both sides by \( y - 3 \):

\[ x = \frac{2y + 4}{y - 3} \]

### Step 5: Expressing \( g(x) \)

Since we have expressed \( x \) in terms of \( y \), we can write the inverse function \( g(x) \):

\[ g(x) = \frac{2x + 4}{x - 3}, \quad x \neq 3 \]

### Step 6: Conclusion

Thus, the function \( g(x) \) is:

\[ g(x) = \frac{2x + 4}{x - 3}, \quad x \neq 3 \]

### Verification

To verify that \( (g \circ f)(x) = x \) and \( (f \circ g)(x) = x \):

1. **Calculate \( g(f(x)) \)**:
\[ g(f(x)) = g\left(\frac{3x + 4}{x - 2}\right) = \frac{2\left(\frac{3x + 4}{x - 2}\right) + 4}{\left(\frac{3x + 4}{x - 2}\right) - 3} \]
Simplifying this will yield \( x \).

2. **Calculate \( f(g(x)) \)**:
\[ f(g(x)) = f\left(\frac{2x + 4}{x - 3}\right) = \frac{3\left(\frac{2x + 4}{x - 3}\right) + 4}{\left(\frac{2x + 4}{x - 3}\right) - 2} \]
Simplifying this will also yield \( x \).