For the given function $f(x) = -5(4)^x - 4$, we analyze the end behavior by considering the limits as $x$ approaches $-\infty$ and $\infty$.

As $x \to -\infty$:

• When $x \to -\infty$, the term $(4)^x \to 0$ because $4^x$ approaches 0 as $x$ becomes more negative.
• Therefore, $f(x) \approx -5(0) - 4 = -4$.

Thus: $\text{As } x \to -\infty, f(x) \to -4$

As $x \to \infty$:

• When $x \to \infty$, the term $(4)^x \to \infty$ because exponential functions grow rapidly for positive large $x$.
• This means $f(x) \to -5(\infty) - 4 = -\infty$.

Thus: $\text{As } x \to \infty, f(x) \to -\infty$

Summary:

• As $x \to -\infty$, $f(x) \to -4$.
• As $x \to \infty$, $f(x) \to -\infty$.

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Do you want further details or have any questions?

Here are 5 related questions for deeper understanding:

1. What is the horizontal asymptote of $f(x)$?
2. How does changing the base of the exponential function affect end behavior?
3. How would the end behavior change if the coefficient of $(4)^x$ were positive?
4. What happens to the graph if you shift $f(x)$ vertically or horizontally?
5. Can you describe the behavior of the function for small values of $x$?

Tip: When analyzing end behavior, focus on how the exponential part dominates for large values of $x$.