Let's analyze the question.

The function $g(x) = -f\left(-\frac{1}{5}x\right) - 4$ modifies $f(x)$ in several ways:

1. Horizontal Scaling and Reflection: The input to $f$ is scaled by $-\frac{1}{5}$, causing a horizontal stretch by a factor of 5 and a reflection across the y-axis.
2. Vertical Reflection: The output of $f$ is negated, reflecting $f(x)$ vertically.
3. Vertical Shift: The function is shifted downward by 4 units.

Domain of $g(x)$

1. The domain of $f(x)$ is $(-\infty, 2]$.
2. Substituting $-\frac{1}{5}x$ as the input means solving for $x$ when $-\frac{1}{5}x \leq 2$: $-\frac{1}{5}x \leq 2 \implies x \geq -10$ Therefore, the domain of $g(x)$ is $[ -10, \infty )$.

Range of $g(x)$

1. The range of $f(x)$ is $[-13, \infty)$.
2. Applying the vertical reflection via $-f(x)$, the range becomes $(-\infty, 13]$.
3. Shifting the function downward by 4 units transforms the range to: $(-\infty, 13 - 4] = (-\infty, 9]$

Final Answer

• Domain: $[ -10, \infty )$
• Range: $(-\infty, 9]$

Let me know if you'd like further explanation or need help with related questions!

Here are some related questions you might consider:

1. How do transformations like reflection or scaling affect the domain and range of a function?
2. What happens to the domain if $f(x)$ includes restrictions like square roots or logarithms?
3. How would a horizontal translation affect the function $g(x)$?
4. If $f(x)$ were a specific function (e.g., a parabola), how would its transformations appear graphically?
5. How can you verify transformations of functions algebraically and graphically?

Tip: Always analyze transformations in steps to systematically understand their effects on the domain and range.