Which expressions describe the end behaviors of the function, $f(x)=\left(\frac{1}{3}\right)^x-4$, modeled by the graph?

Select each correct answer.

As $x$ decreases without bound, $f(x)$ approaches the line $y=4$.
As $x$ decreases without bound, $f(x)$ increases without bound.
As $x$ increases without bound, $f(x)$ approaches the line $y=-4$.
As $x$ decreases without bound, $f(x)$ approaches zero.

Question

03-11-2026
Mathematics

Answered

Which expressions describe the end behaviors of the function, $f(x)=\left(\frac{1}{3}\right)^x-4$, modeled by the graph?

Select each correct answer.

As $x$ decreases without bound, $f(x)$ approaches the line $y=4$.
As $x$ decreases without bound, $f(x)$ increases without bound.
As $x$ increases without bound, $f(x)$ approaches the line $y=-4$.
As $x$ decreases without bound, $f(x)$ approaches zero.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T20:17:56-05:00

Analyze the function $f(x) = \left(\frac{1}{3}\right)^x - 4$, which consists of an exponential decay term shifted down by 4.
Observe that as $x \to \infty$, $\left(\frac{1}{3}\right)^x \to 0$, so the function approaches $-4$.
Recognize that as $x \to -\infty$, $\left(\frac{1}{3}\right)^x \to \infty$, meaning the function increases without bound.
The correct statements are: As $x$ decreases without bound, $f(x)$ increases without bound; and as $x$ increases without bound, $f(x)$ approaches the line $y = -4$.

Explanation

Analyzing the Function Structure

To understand the behavior of the function $f(x) = \left(\frac{1}{3}\right)^x - 4$, we examine its values as $x$ becomes very large (increases without bound) and very small (decreases without bound, or approaches $-\infty$). This function is an exponential function shifted vertically by $-4$. The term $\left(\frac{1}{3}\right)^x$ is a basic exponential decay function because the base $0 < \frac{1}{3} < 1$.

End Behavior as $x$ Increases Without Bound

As $x$ increases without bound ($x \to \infty$), the term $\left(\frac{1}{3}\right)^x$ becomes $\frac{1}{3^x}$. As $x$ grows, $3^x$ grows very large, so the fraction $\frac{1}{3^x}$ approaches zero. Therefore, the function $f(x) = \left(\frac{1}{3}\right)^x - 4$ approaches $0 - 4 = -4$. This means the graph has a horizontal asymptote at $y = -4$. Thus, as $x$ increases without bound, $f(x)$ approaches the line $y = -4$.

End Behavior as $x$ Decreases Without Bound

As $x$ decreases without bound ($x \to -\infty$), we can evaluate the function by setting $x = -u$, where $u$ is a large positive value. Substituting this into the function, we get:
\[f(-u) = \left(\frac{1}{3}\right)^{-u} - 4 = 3^u - 4\]
As $u$ increases without bound, $3^u$ also increases without bound. Subtracting 4 from a value that is growing toward infinity does not change the fact that it is still increasing without bound. Therefore, as $x$ decreases without bound, $f(x)$ increases without bound.

Selecting the Correct Statements

Based on our analysis:

As $x$ decreases without bound, $f(x)$ increases without bound.
As $x$ increases without bound, $f(x)$ approaches the line $y = -4$.

Comparing these to the options provided, the two correct statements are the second and third ones.

Examples

In biology, exponential decay models are often used to describe how a drug is eliminated from the bloodstream over time. If $f(x)$ represents the concentration of a medication in the blood after $x$ hours, the vertical shift (like -4 in our example) might represent a baseline concentration level that the drug approaches as it clears. Understanding the limits of such functions allows medical professionals to predict long-term drug concentration levels in a patient's body.