The given function in the image is:

$f(x) = \left(\frac{1}{5}\right)^x$

The problem states that this function is translated up 4 units. When translating a function vertically, we add or subtract from the entire function. In this case, adding 4 units translates the function upwards.

The translated function becomes:

$g(x) = \left(\frac{1}{5}\right)^x + 4$

From the options in the image, the correct equation is:

$g(x) = \left(\frac{1}{5}\right)^x + 4$

This matches the third option.

Would you like a more detailed explanation? Here are some related questions to further your understanding:

1. What would happen if the function was translated down 4 units?
2. How would you translate the function horizontally, say by shifting it left or right?
3. How does adding a constant affect the graph of an exponential function?
4. How would you write the equation if the function was stretched or compressed vertically?
5. What would the equation look like if it was reflected over the x-axis?

Tip: When translating a function vertically, always add or subtract the desired value directly to the function itself.