Problem 3 - Elementary and Intermediate Algebra

Question

The graph of \(f(x)=3^{x}\) approaches, but never touches, the negative portion of the \(x\) -axis. Thus, the \(x\) -axis is an ___________ of the graph.

Step-by-Step Solution

Answer

The x-axis is a horizontal asymptote of the graph.

Step 1: Understand the Concept of an Asymptote

An asymptote is a line that a graph approaches as it goes towards infinity. In the case of vertical or horizontal asymptotes, the function comes arbitrarily close to the line but does not actually meet it.

Step 2: Analyze the Function

Consider the function given, which is exponential: \( f(x) = 3^x \). As \( x \) approaches negative infinity, the value of \( 3^x \) approaches zero, meaning the graph gets closer to the x-axis but never touches it.

Step 3: Identify the Asymptote Type

For the function \( f(x) = 3^x \), as \( x \to -\infty \), \( f(x) \to 0 \). The graph is approaching the line \( y = 0 \), which is the x-axis. This means the x-axis is a horizontal asymptote.

Key Concepts

Exponential Functions

Exponential functions are an important concept in mathematics, characterized by having a variable as an exponent. Such functions take the general form of \( f(x) = a^{x} \), where \( a \) is a positive real number. In the case of our example, \( a = 3 \). This function grows at an exponential rate as \( x \) increases. Unlike linear functions, which increase steadily, exponential functions can spike up quickly as the value of \( x \) increases.
When \( x \) is a positive number, the function outputs larger values. Conversely, when \( x \) is negative, the function produces fractional values between 0 and 1. This behavior is key to understanding the graph's interaction with the axes.

• Exponential growth: rapidly increasing function
• Base \( a \) determines the rate of growth

The exponential function is fascinating because it captures growth processes in real-life scenarios, such as population growth or compound interest.

Graph Analysis

Analyzing the graph of an exponential function involves looking at how it behaves as the input variable changes. For \( f(x) = 3^x \), the graph forms a smooth curve.
It rises steeply as \( x \) increases to the right, displaying the characteristic exponential growth. As \( x \) becomes very large in the positive direction, the graph grows rapidly without any bound.
In contrast, when \( x \) becomes very large in the negative direction, the graph approaches the x-axis but never quite touches it. This is due to the nature of exponential decay in this region.

• Quick rise as \( x \) becomes more positive
• Approaches but never reaches the x-axis as \( x \) becomes more negative

The key to graph analysis is observing how the graph positions itself with respect to the axes, providing insights into the function's behavior throughout its domain.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but may never actually touch. For the function \( f(x) = 3^x \), the x-axis acts as a horizontal asymptote.
As \( x \) moves towards negative infinity, the value of \( 3^x \) decreases and approaches zero, but it never equals zero. This indicates that the x-axis, or the line \( y = 0 \), is a horizontal asymptote. This distinction helps us understand where the graph levels off as \( x \) spans various magnitudes.

• X-axis as \( y = 0 \) is the horizontal asymptote for \( f(x) = 3^x \)
• The function gets closer to 0 as \( x \) becomes increasingly negative

A simple horizontal line may seem insignificant, but recognizing these asymptotic behaviors is crucial in analyzing the long-term behavior of functions.