Thank you for the clarification. The function you've identified, $f(x) = \left(\frac{1}{5}\right)^x - 1$, corresponds to an exponential function that has undergone a transformation.

Key Characteristics of the Correct Function:

1. Exponential Decay: The base $\frac{1}{5}$ is between 0 and 1, which causes the graph to decay as $x$ increases.
2. Vertical Shift: The subtraction of 1 from the function $\left(\frac{1}{5}\right)^x$ shifts the graph downward by 1 unit.
3. Asymptote: The horizontal asymptote of the function is at $y = -1$, as the function approaches but never reaches this value for large $x$.

Equation:

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

This means the graph represents an exponential decay function that has been shifted down by 1 unit.

Would you like any further details or explanations?

Here are some follow-up questions:

1. How do the base values of an exponential function (greater than 1 vs. between 0 and 1) affect the graph?
2. What is the effect of vertical shifts on the horizontal asymptote of an exponential function?
3. How does the function $\left(\frac{1}{5}\right)^x$ differ from the function $\left(\frac{5}{1}\right)^x$ in terms of graph behavior?
4. What are the general rules for graphing transformations such as shifts and reflections?
5. How would the graph change if the equation were $f(x) = \left(\frac{1}{5}\right)^x + 1$?

Tip: When analyzing exponential functions, always check for any shifts that modify the asymptote, as these are key to identifying the correct equation.