Problem 31 - College Algebra

Question

For the following exercises, describe the end behavior of the graphs of the functions. $$ f(x)=3(4)^{-x}+2 $$

Step-by-Step Solution

Answer

As \(x\to+\infty\), \(f(x)\to2\); as \(x\to-\infty\), \(f(x)\to+\infty\).

Step 1: Analyze the Function

The function is given by \( f(x) = 3(4)^{-x} + 2 \). This is an exponential decay function because the base of the exponential term, \(4^{-x}\), represents decay as \(x\) increases. The constant \(+2\) shifts the graph vertically.

Step 2: Determine End Behavior as \(x \to +\infty\)

As \(x\) approaches positive infinity, the term \(3(4)^{-x}\) approaches zero because \((4)^{-x} \) tends towards zero. Thus, the function approaches \(f(x) = 2 \). Therefore, the end behavior as \(x \to +\infty\) is that \(f(x) \to 2\).

Step 3: Determine End Behavior as \(x \to -\infty\)

As \(x\) approaches negative infinity, the term \(4^{-x}\) grows very large because a negative exponent becomes a large positive exponent when the negative sign is removed. Consequently, \(3(4)^{-x} \) also becomes very large, and \(f(x)\) approaches infinity. Thus, as \(x \to -\infty\), \(f(x) \to +\infty\).

Key Concepts

Understanding End Behavior in Exponential Functions

End behavior in mathematics refers to the behavior of a graph as it approaches positive or negative infinity. It's important to understand this concept when analyzing exponential functions. In the case of the function \( f(x) = 3(4)^{-x} + 2 \), we observe two specific types of end behavior.
As \( x \to +fty \), or as \( x \) becomes a very large positive number, \( 4^{-x} \) tends towards zero. Since this term is multiplied by 3, the entire term \( 3(4)^{-x} \) also approaches zero. The remaining part of the function, due to the constant \(+ 2\), means \( f(x) \to 2 \).
On the other hand, as \( x \to -fty \), or as \( x \) becomes a very large negative number, the exponent negates the negative sign making it positive. Thus, \( 4^{-x} \) grows large, and correspondingly \( 3(4)^{-x} \) becomes large. Here, \( f(x) \to +fty \). Understanding these two behaviors helps analyze the overall graph and makes it easier to predict the function's long-term trends.

Exploring Exponential Decay

Exponential decay is a process where quantities decrease rapidly at first and then slowly over time. This is noticeable in our function \( f(x) = 3(4)^{-x} + 2 \), which models exponential decay.
When the base of an exponential function is between 0 and 1, it depicts decay. Here, the term \((4)^{-x}\) indicates such behavior, as it results in a fraction less than one for positive \(x\). The more \(x\) increases, the smaller \((4)^{-x} \) becomes, reducing the overall value of the term \(3(4)^{-x}\).

• This gradual decrease is prominent as you move to the right on the x-axis.
• It leads the graph to approach its horizontal asymptote as \(x \to +fty\), which is reflected by the constant term \(+ 2\).

Exponential decay functions like this one are often seen in real-world scenarios like radioactive decay or depreciation of asset value.

Graph Analysis of Exponential Functions

Graph analysis begins by identifying key features of a function: intercepts, asymptotes, and overall shape. Let’s examine the graph for \( f(x) = 3(4)^{-x} + 2 \).
First, recognize the horizontal asymptote. Here, it's \(y = 2\), which the graph approaches but never quite touches as \(x \to +fty\).
The y-intercept, the point where the graph crosses the y-axis, is attained when \(x = 0\). Plugging zero into our function gives \(f(0) = 3 \cdot (4)^{0} + 2 = 5\). Thus, the y-intercept is at the point \((0, 5)\).

• The exponential decay results in a curve that slopes downward from left (as \(x\) decreases) to the horizontal asymptote on the right.
• This graph visually embodies the function’s end behaviors and its decay characteristics.

To fully grasp the graph's intricacies, it is beneficial to plot several \(x\) values to see how the function behaves between these major points.