The function $$ f $$ is given by $$ f(x)=4 \cdot 3^{(z-2)}+1 $$. The function $$ g $$ is given by $$ g(x)=f^{-1}(x) $$. For which of the following values does $$ g(x)=-3 x $$ ?

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To find the values for which $$ g(x) = -3x $$, we first need to determine the inverse function $$ g(x) = f^{-1}(x) $$ of the function $$ f(x) = 4 \cdot 3^{(x-2)} + 1 $$.

### Step 1: Find the inverse function $$ f^{-1}(x) $$

1. Start with the equation for $$ f(x) $$:
   $$
   y = 4 \cdot 3^{(x-2)} + 1
   $$

2. Rearrange the equation to solve for $$ x $$:
   $$
   y - 1 = 4 \cdot 3^{(x-2)}
   $$
   $$
   \frac{y - 1}{4} = 3^{(x-2)}
   $$

3. Take the logarithm of both sides:
   $$
   \log_3\left(\frac{y - 1}{4}\right) = x - 2
   $$

4. Solve for $$ x $$:
   $$
   x = \log_3\left(\frac{y - 1}{4}\right) + 2
   $$

Thus, the inverse function is:
$$
f^{-1}(x) = \log_3\left(\frac{x - 1}{4}\right) + 2
$$

### Step 2: Set up the equation $$ g(x) = -3x $$

Now we set $$ g(x) = f^{-1}(x) $$ equal to $$ -3x $$:
$$
\log_3\left(\frac{x - 1}{4}\right) + 2 = -3x
$$

### Step 3: Solve the equation

1. Rearranging gives:
   $$
   \log_3\left(\frac{x - 1}{4}\right) = -3x - 2
   $$

2. Exponentiating both sides to eliminate the logarithm:
   $$
   \frac{x - 1}{4} = 3^{-3x - 2}
   $$

3. Multiply both sides by 4:
   $$
   x - 1 = 4 \cdot 3^{-3x - 2}
   $$
   $$
   x - 1 = \frac{4}{9} \cdot 3^{-3x}
   $$

4. Rearranging gives:
   $$
   x - 1 = \frac{4}{9} \cdot \frac{1}{3^{3x}}
   $$

### Step 4: Solve for $$ x $$

This equation is complex and may require numerical methods or graphing to find solutions. However, we can also check specific values of $$ x $$ to see if they satisfy the equation.

Let's check some integer values for $$ x $$:

- For $$ x = 0 $$:
  $$
  0 - 1 = -1 \quad \text{and} \quad \frac{4}{9} \cdot 3^{0} = \frac{4}{9} \quad \text{(not equal)}
  $$

- For $$ x = 1 $$:
  $$
  1 - 1 = 0 \quad \text{and} \quad \frac{4}{9} \cdot 3^{-3} = \frac{4}{9} \cdot \frac{1}{27} = \frac{4}{243} \quad \text{(not equal)}
  $$

- For $$ x = -1 $$:
  $$
  -1 - 1 = -2 \quad \text{and} \quad \frac{4}{9} \cdot 3^{3} = \frac{4}{9} \cdot 27 = 12 \quad \text{(not equal)}
  $$

- For $$ x = -2 $$:
  $$
  -2 - 1 = -3 \quad \text{and} \quad \frac{4}{9} \cdot 3^{6} = \frac{4}{9} \cdot 729 = 324 \quad \text{(not equal)}
  $$

Continuing this process or using numerical methods will yield the specific values of $$ x $$ for which $$ g(x) = -3x $$.

### Conclusion

To find the exact values, numerical methods or graphing tools may be necessary. Would you like me to perform a numerical solution for this equation?
To find the values of $$ x $$ where $$ g(x) = -3x $$, first find the inverse function $$ g(x) = f^{-1}(x) $$ of $$ f(x) = 4 \cdot 3^{(x-2)} + 1 $$. Setting $$ g(x) = -3x $$ leads to a complex equation that may require numerical methods to solve. After testing integer values, no exact solutions are found within the tested range.

The function \( f \) is given by \( f(x)=4 \cdot 3^{(z-2)}+1 \). The function \( g \) is given by \( g(x)=f^{-1}(x) \). For which of the following values does \( g(x)=-3 x \) ?

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