Problem 18 - Precalculus Mathematics for Calculus

Question

Graph both functions on one set of axes. $$f(x)=\left(\frac{2}{3}\right)^{x} \quad \text { and } \quad g(x)=\left(\frac{4}{3}\right)^{x}$$

Step-by-Step Solution

Answer

Plot points for both functions and connect with smooth curves; observe decay for \(f(x)\) and growth for \(g(x)\).

Step 1: Understand the Functions

First, identify the type of functions we have. Both \(f(x) = \left(\frac{2}{3}\right)^x\) and \(g(x) = \left(\frac{4}{3}\right)^x\) are exponential functions. Here, the base \(\frac{2}{3} < 1\) indicates a decreasing exponential function, while \(\frac{4}{3} > 1\) indicates an increasing exponential function.

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

To graph \(f(x) = \left(\frac{2}{3}\right)^x\), calculate its value for various \(x\) values. For example:- \(x = -1\), \(f(-1) = \left(\frac{3}{2}\right)^1 = 1.5\)- \(x = 0\), \(f(0) = 1\)- \(x = 1\), \(f(1) = \frac{2}{3}\)- \(x = 2\), \(f(2) = \left(\frac{2}{3}\right)^2 = \frac{4}{9}\)Create a table with these values.

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

Similarly, calculate values for \(g(x) = \left(\frac{4}{3}\right)^x\):- \(x = -1\), \(g(-1) = \left(\frac{3}{4}\right)^1 = 0.75\)- \(x = 0\), \(g(0) = 1\)- \(x = 1\), \(g(1) = \frac{4}{3}\)- \(x = 2\), \(g(2) = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \approx 1.78\)Add these values to another table.

Step 4: Set Up the Axes

Draw a coordinate system with the x-axis representing a range of \(x\) values and the y-axis representing the function values. Mark the points calculated for both functions.

Step 5: Plot the Points for f(x) and g(x)

On the same graph, plot the points for \(f(x)\) using its values from Step 2 and connect them with a smooth curve to represent the exponential decrease. Then, plot the points for \(g(x)\) using the values from Step 3 and connect them with a smooth curve to represent the exponential growth.

Step 6: Check for Intersection and Behavior

Finally, observe the behavior of the curves. For \(f(x)\), the curve should approach the x-axis as \(x\) increases, showing exponential decay. For \(g(x)\), the curve increases as \(x\) increases, demonstrating exponential growth. The graphs will intersect at the y-axis where both functions equal 1 at \(x = 0\).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions involves plotting the set of all points that satisfy an exponential equation on the coordinate system. Each exponential function has a distinct shape based on its base. For a given function of the form \(f(x) = a^x\), where \(a\) is a positive constant:

• If \(a > 1\), the function exhibits exponential growth and the graph slopes upwards as \(x\) increases.
• If \(0 < a < 1\), the function shows exponential decay, and the graph slopes downwards as \(x\) increases.

When graphing, it is essential to

• Choose a range of \(x\) values, including negative and positive integers, to see the behavior across the axes.
• Calculate the corresponding \(y\) values for each \(x\).
• Plot these points on the coordinate plane and connect them smoothly.

For the functions \(f(x) = \left(\frac{2}{3}\right)^x\) and \(g(x) = \left(\frac{4}{3}\right)^x\), observe how their graphs diverge, with \(f(x)\) decreasing and \(g(x)\) increasing, demonstrating their distinct exponential behaviors.

Exponential Growth and Decay

Exponential functions are vital in modeling various real-world phenomena, with each function showcasing either growth or decay. Understanding these concepts is crucial:**Exponential Growth** occurs when a quantity increases multiplicatively over time. The base of the exponential function \(a\) is greater than 1. A few examples include:

• Population growth
• Interest compounding
• Biological processes like bacterial growth

As an example, the function \(g(x) = \left(\frac{4}{3}\right)^x\) represents exponential growth. The graph rises steeply, indicating a rapid increase in value as \(x\) becomes larger.**Exponential Decay**, on the other hand, happens when quantities decrease over time. The base \(a\) of an exponential decay function is between 0 and 1. Examples include:

• Radioactive decay
• Cooling temperatures
• Depreciation of assets

For \(f(x) = \left(\frac{2}{3}\right)^x\), it models exponential decay, and its graph steadily approaches the x-axis, reflecting diminishing values as \(x\) increases.

Coordinate System

In mathematics, the coordinate system is an essential framework used for graphically representing equations and functions, including exponential ones. The Cartesian coordinate system comprises two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). They intersect at a point known as the origin, denoted by (0, 0).**Key Features:**

• The x-axis captures the independent variable (often \(x\)) values.
• The y-axis holds the dependent variable (often \(y\) or function value) values.
• The plane created is divided into four quadrants, allowing both positive and negative values for x and y.

When graphing functions like \(f(x)\) and \(g(x)\),

• Select appropriate scale and range for both x and y to ensure visibility of the graph behaviors.
• Regularly label the axes to provide context and clarity for the graph interpretation.
• Through plotting the points calculated from each function, the coordinate system helps visualize how exponential functions grow or decay over the domain.