Problem 17 - Precalculus Mathematics for Calculus

Question

Graph both functions on one set of axes. $$f(x)=4^{x} \quad \text { and } \quad g(x)=7^{x}$$

Step-by-Step Solution

Answer

Graph both functions on the same set of axes; note that \( g(x) = 7^x \) rises faster than \( f(x) = 4^x \).

Step 1: Understand the Functions

We have two functions to graph: \( f(x) = 4^x \) and \( g(x) = 7^x \). Both are exponential functions where the base is greater than 1, indicating they will both be increasing functions.

Step 2: Identify Key Points

For \( f(x) = 4^x \) and \( g(x) = 7^x \):- At \( x = 0 \), \( f(0) = 4^0 = 1 \) and \( g(0) = 7^0 = 1 \).- At \( x = 1 \), \( f(1) = 4^1 = 4 \) and \( g(1) = 7^1 = 7 \).- At \( x = -1 \), \( f(-1) = 4^{-1} = \frac{1}{4} \) and \( g(-1) = 7^{-1} = \frac{1}{7} \).

Step 3: Plot the Key Points

On a set of axes, plot the points for each function:- For \( f(x) \): \( (0,1), (1,4), (-1,\frac{1}{4}) \).- For \( g(x) \): \( (0,1), (1,7), (-1,\frac{1}{7}) \).Each set of points will serve as a guide for drawing the curves.

Step 4: Draw the Exponential Curves

Using the plotted points for \( f(x) \) and \( g(x) \), draw smooth curves through the points:- \( f(x) = 4^x \) will increase moderately, showing steeper curves than linear functions.- \( g(x) = 7^x \) will rise more steeply than \( f(x) \) because of the larger base.

Step 5: Analyze the Graph

On the graph, notice:- Both functions intersect the y-axis at the same point (0,1).- \( g(x) \) (in blue) grows faster than \( f(x) \) (in red) as shown by the greater y-values at positive x.- Both decrease toward the x-axis in the negative x-direction, but \( g(x) \) approaches the axis more slowly.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions involves plotting points on a coordinate system to illustrate how exponential equations behave. These types of functions are expressed in the form \( f(x) = a^x \), where \( a \) is a positive constant. When graphing, it is essential to identify key points such as where the graph crosses the y-axis and how the function behaves as \( x \) becomes positive or negative.
\( f(x) = 4^x \) and \( g(x) = 7^x \) are both functions with their own unique curves, showing exponential increases. On a graph, you would typically:

• Mark the y-intercept at \( (0,1) \) since \( a^0 = 1 \) for any base \( a \) greater than zero.
• Plot a few points for positive and negative values of \( x \) to define the curve. As these values rise, you'll see an increasingly steeper curve due to the nature of exponential growth.
• Draw a smooth curve through the points to reflect the continuous and increasing nature of the exponential function.

Observing these plotted graphs helps one see how quickly functions diverge from each other based on their bases.

Base of Exponential Functions

The base of an exponential function significantly affects the graph's shape and its rate of growth or decay. In exponential functions, such as \( f(x) = 4^x \) and \( g(x) = 7^x \), the base is the number that is raised to the power of \( x \).
A crucial point to remember is:

• If the base \( a > 1 \), the function experiences exponential growth. As the base increases, the function grows faster.
• If the base \( 0 < a < 1 \), the function would instead represent exponential decay, decreasing as \( x \) increases.

For our functions:

• \( f(x) = 4^x \) shows a moderate rate of increase with a base of 4.
• \( g(x) = 7^x \) shows a faster rate of increase due to its larger base of 7.

Understanding the base of an exponential function is crucial to predicting and controlling how a function will behave under different conditions.

Exponential Growth

Exponential growth refers to a rapid increase in the value of a function as \( x \) becomes larger. It is characterized by a constant percentage rate of growth, often described in the context of populations, finance, and natural phenomena. In the case of exponential functions like \( f(x) = 4^x \) and \( g(x) = 7^x \), the concept of exponential growth plays a crucial role.
Here are the fundamental aspects of exponential growth:

• Both functions \( f(x) \) and \( g(x) \) show growth because their bases are greater than 1.
• Exponential growth is evident in the steep upward curve of the graphs as \( x \) increases.
• The difference in growth rates between \( f(x) \) and \( g(x) \) becomes more apparent over larger values of \( x \), where \( g(x) \) outpaces \( f(x) \) significantly.

This behavior is a hallmark of how exponential functions operate, providing valuable insight into the power of exponential changes and their impacts across different fields and applications.