Problem 15 - Precalculus Mathematics for Calculus

Question

Graph both functions on one set of axes. $$f(x)=2^{x} \quad \text { and } \quad g(x)=2^{-x}$$

Step-by-Step Solution

Answer

Graph both functions on the same axes, showing exponential growth for \( f(x) \) and decay for \( g(x) \).

Step 1: Identify the Functions

We need to graph two functions: 1. \( f(x) = 2^{x} \) which is an exponential growth function.2. \( g(x) = 2^{-x} \) which is an exponential decay function.

Step 2: Create a Table of Values for f(x)

Select a range of \( x \) values to evaluate for \( f(x) = 2^x \). For example:- \( x = -2 \): \( f(-2) = \frac{1}{4} \)- \( x = -1 \): \( f(-1) = \frac{1}{2} \)- \( x = 0 \): \( f(0) = 1 \)- \( x = 1 \): \( f(1) = 2 \)- \( x = 2 \): \( f(2) = 4 \)This will help us understand the shape of \( f(x) \).

Step 3: Create a Table of Values for g(x)

Select a range of \( x \) values to evaluate for \( g(x) = 2^{-x} \). For example:- \( x = -2 \): \( g(-2) = 4 \)- \( x = -1 \): \( g(-1) = 2 \)- \( x = 0 \): \( g(0) = 1 \)- \( x = 1 \): \( g(1) = \frac{1}{2} \)- \( x = 2 \): \( g(2) = \frac{1}{4} \)This will help us understand the shape of \( g(x) \).

Step 4: Plot the Points on a Graph

Draw a set of axes with labeled units. Then plot each calculated point for \( f(x) \) and \( g(x) \). Join the points smoothly for each function to resemble their exponential nature.

Step 5: Analyze the Graphs

\( f(x) = 2^x \) shows exponential growth, meaning it rises sharply as \( x \) increases. Conversely, \( g(x) = 2^{-x} \) declines sharply as \( x \) increases, showing exponential decay. Both functions intersect at the point \((0,1)\).

Key Concepts

Understanding Exponential Growth

Exponential growth occurs when the rate of increase of a quantity is proportional to its current value. This can be seen in functions like \( f(x) = 2^x \), where the base greater than 1 indicates continuous growth.
If you plug into this function, you'll notice the outputs increase rapidly:

• At \(x = -2\), \( f(x) = \frac{1}{4} \), for a low initial value.
• But when \(x = 2\), \( f(x) = 4 \) shows rapid growth.

Each step forward on the \(x\)-axis multiplies the output by 2. As \(x\) increases, the graph takes on a sharp upward curve. This behavior is characteristic of exponential growth and is crucial in fields like population dynamics and compound interest calculations.

Exploring Exponential Decay

Exponential decay describes situations where a quantity decreases at a rate proportional to its current value. Such behavior is observed in functions like \( g(x) = 2^{-x} \), where the negative exponent leads to halving the output with every step forward on the \(x\)-axis.
Here's how it behaves at different \(x\) values:

• At \(x = -2\), \( g(x) = 4 \) indicates a higher initial value.
• As \(x = 2\), \( g(x) = \frac{1}{4} \) shows a decrease.

The graph for this function gives a downward curve, decreasing sharply as \(x\) becomes larger. Fields like radioactive decay and depreciation of asset values commonly involve this type of function. Understanding exponential decay is essential in these contexts, as it helps predict how quickly a quantity will diminish over time.

Graphing Exponential Functions

Graphing functions like \( f(x) = 2^x \) and \( g(x) = 2^{-x} \) together allows for a visual comparison of exponential growth and decay on the same axes. To create these graphs, follow these steps:
- Draw axes with equally spaced intervals.
- Plot values such as \((-2, \frac{1}{4})\), \((0, 1)\), and \((2, 4)\) for \( f(x) \).
- For \( g(x) \), include points like \((-2, 4)\), \((0, 1)\), and \((2, \frac{1}{4})\).
The intersection at \((0, 1)\) reflects the point where both function values equal, providing a reference point.
This exercise demonstrates how exponential growth travels upwards steeply and decay plummets downwards equally fast. Recognizing these patterns visually helps solidify the mathematical concepts and gives insight into how real-world phenomena unfold.