Problem 30 - College Algebra

Question

For the following exercises, describe the end behavior of the graphs of the functions. $$ f(x)=3\left(\frac{1}{2}\right)^{x}-2 $$

Step-by-Step Solution

Answer

As \( x \to \infty \), \( f(x) \to -2 \); as \( x \to -\infty \), \( f(x) \to \infty \).

Step 1: Identify the Type of Function

The given function is \( f(x) = 3\left(\frac{1}{2}\right)^{x} - 2 \). This is an exponential function because it has the form \( a \cdot b^{x} + c \) where \( a = 3 \), \( b = \frac{1}{2} \), and \( c = -2 \). Since \( b = \frac{1}{2} \) is between 0 and 1, this is a decaying exponential function.

Step 2: Determine End Behavior as x Approaches Infinity

For an exponential decay function of the form \( a \cdot b^{x} + c \) with \( 0 < b < 1 \), as \( x \to \infty \), \( b^{x} \to 0 \). Thus the function approaches \( c \). So, as \( x \to \infty \), \( f(x) = 3\left(\frac{1}{2}\right)^{x} - 2 \to -2 \).

Step 3: Determine End Behavior as x Approaches Negative Infinity

As \( x \to -\infty \), for an exponential function \( b^{x} \) where \( 0 < b < 1 \), \( b^{x} \to \infty \). Therefore, \( 3\left(\frac{1}{2}\right)^{x} \to \infty \) and so \( f(x) = 3\left(\frac{1}{2}\right)^{x} - 2 \to \infty \).

Step 4: Summarize the End Behavior

The end behavior of the function \( f(x) = 3\left(\frac{1}{2}\right)^{x} - 2 \) is that as \( x \to \infty \), \( f(x) \to -2 \); and as \( x \to -\infty \), \( f(x) \to \infty \). This means the graph approaches -2 from above as \( x \) increases and rises indefinitely as \( x \) decreases.

Key Concepts

Understanding End Behavior

In mathematics, understanding the end behavior of a function is crucial, especially for graph interpretation. The end behavior refers to how the function behaves as the input values become very large or very small. For an exponential decay function, end behavior analysis helps predict how the function behaves when the input, or x-values, approach infinity or negative infinity.
For an exponential decay function like \( f(x)=3\left(\frac{1}{2}\right)^{x}-2 \), as we examine `end behavior`, it's essential to focus on the base, \( b \), of the exponent. Here, \( b = \frac{1}{2} \), which is positive but less than 1, indicating a decreasing exponential function. Therefore,

• As \( x \to \infty \), the exponential term diminishes towards zero, meaning \( f(x) \) approaches the constant \( c \), which is -2.
• As \( x \to -\infty \), the behavior diverges as the exponent makes the base's effect amplify, causing \( f(x) \) to grow infinitely.

Recognize that understanding end behavior allows us to anticipate how graphs look over a long stretch of the input axis.

Approaching Negative Infinity

Let's dive into negative infinity and its effects on exponential decay functions. When \( x \to -\infty \) in functions like \( f(x)=3\left(\frac{1}{2}\right)^{x}-2 \), something fascinating happens. Consider \( \left(\frac{1}{2}\right)^{-x} \). It effectively flips the fraction, causing an increase rather than a decrease.

As this negative exponent increases without bound, it makes the entire term \( 3\left(\frac{1}{2}\right)^{x} \) grow rapidly. Since we are subtracting 2, \( f(x) \) shoots upwards towards infinity. This behavior explains the left side of the graph where it seems to rise sharply, defying horizontal limits. It's critical to grasp how negative exponents cause exponential growth in decay functions when extending leftward on a graph.

Reaching Positive Infinity

Analyzing how exponential decay functions behave as \( x \to \infty \) is different from negative infinity dynamics. For decay functions like \( f(x)=3\left(\frac{1}{2}\right)^{x}-2 \):

• Since \( \frac{1}{2} \) is less than 1, each increment in x makes \( \left(\frac{1}{2}\right)^{x} \) smaller.
• This term becomes almost negligible, practically approaching zero as x becomes extremely large.

Thus, the function gravitates towards the horizontal line \( y = c = -2 \). The graph levels towards this line, showing an asymptotic approach. This approach means the function will get closer to -2 but never quite touch or cross it. Understanding this lets us predict how the graph flattens at the higher reaches of the x-axis.

Graph Analysis of Exponential Decay

Interpreting the graph of an exponential decay function provides insight into the function's end behavior. For the function \( f(x) = 3\left(\frac{1}{2}\right)^{x} - 2 \), the graph showcases how exponential decay is mirrored in its asymptotic trends.

When we plot this function:

• The graph starts high on the left near negative infinity and descends towards the horizontal asymptote \( y = -2 \) on the right.
• It depicts a quick drop initially, flattens out around the asymptote, and aligns closely, illustrating how decay works practically.

Understanding such visual cues enhances comprehension, offering a real-world glimpse into how values shift and stabilize. This graphic analysis communicates not just numbers but a tangible story of change that occurs in decay functions.