Questions: Describe the end behavior of the graph of the function f(x)=5(4)^x-10. As x → -∞, f(x) → As x → ∞, f(x) →

Transcript text: Describe the end behavior of the graph of the function $f(x)=5(4)^{x}-10$. As $x \rightarrow-\infty, f(x) \rightarrow$ $\square$ As $x \rightarrow \infty, f(x) \rightarrow$ $\square$

Solution

Solution Steps

To describe the end behavior of the function $ f(x) = 5(4)^x - 10 $, we need to analyze the behavior of the exponential term $ (4)^x $ as $ x $ approaches $-\infty$ and $\infty$.

As $ x \rightarrow -\infty $, the term $ (4)^x $ approaches 0 because any positive number raised to a negative power approaches zero. Therefore, $ f(x) $ approaches $-10$.
As $ x \rightarrow \infty $, the term $ (4)^x $ grows exponentially large. Therefore, $ f(x) $ approaches infinity.

Step 1: Analyze the Limit as $ x \rightarrow -\infty $

As $ x $ approaches $-\infty$, the term $ (4)^x $ approaches $ 0 $. Therefore, we can evaluate the limit of the function $ f(x) = 5(4)^x - 10 $:

\[ \lim_{x \to -\infty} f(x) = 5 \cdot 0 - 10 = -10 \]

Step 2: Analyze the Limit as $ x \rightarrow \infty $

As $ x $ approaches $ \infty $, the term $ (4)^x $ grows exponentially large. Thus, we can evaluate the limit of the function:

\[ \lim_{x \to \infty} f(x) = 5 \cdot \infty - 10 = \infty \]