Problem 79 - Contemporary Precalculus

Question

(a) Using the viewing window with \(-4 \leq x \leq 4\) and \(-1 \leq y \leq 8, \operatorname{graph} f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=2^{x}\) on the same screen. If you think of the \(y\) -axis as a mirror, how would you describe the relationship between the two graphs? (b) Without graphing, explain how the graphs of \(g(x)=2^{x}\) and \(k(x)=2^{-x}\) are related.

Step-by-Step Solution

Answer

Answer: The graphs of the functions \(f(x) = (\frac{1}{2})^x\) and \(g(x) = 2^x\) are symmetrical about the \(y\)-axis, like reflections in a mirror. Without graphing the functions \(g(x) = 2^x\) and \(k(x) = 2^{-x}\), we can observe that \(k(x)\) is the same as \(f(x)\), so their relationship is also symmetrical about the \(y\)-axis.

Step 1: Graph f(x) and g(x)

First, graph the two functions \(f(x) = (\frac{1}{2})^x\) and \(g(x) = 2^x\) on the same screen with the given viewing window, \(-4 \leq x \leq 4\) and \(-1 \leq y \leq 8\). This can be done using a graphing calculator, online graphing tool, or manually.

Step 2: Describe the relationship between the graphs

Observe the graphs of the two functions and consider the \(y\)-axis as a mirror. They are reflections of each other with respect to the \(y\)-axis, like a person and their reflection in a mirror. In mathematical terms, the graphs are symmetrical about the \(y\)-axis.

Step 3: Explain the relationship between g(x) and k(x) without graphing

We need to relate the functions \(g(x) = 2^x\) and \(k(x) = 2^{-x}\) without graphing them. Notice that the function \(k(x)\) can be rewritten as follows: \(k(x) = 2^{-x} = (\frac{1}{2})^x = f(x)\), which is the same function as \(f(x)\) given in part (a). By comparing \(g(x) = 2^x\) and \(f(x) = (\frac{1}{2})^x\), we know that their graphs are symmetrical about the \(y\)-axis. Since \(k(x) = f(x)\), it can be concluded that the graphs of \(g(x) = 2^x\) and \(k(x) = 2^{-x}\) are also symmetrical about the \(y\)-axis.

Key Concepts

Exponential Functions

Exponential functions are fascinating mathematical entities where the variable, in this case, is the exponent. Functions like

• \( f(x) = \left(\frac{1}{2}\right)^x \)
• \( g(x) = 2^x \)

represent basic exponential functions with different bases. The base of an exponential function determines its growth pattern. The function \( f(x) = \left(\frac{1}{2}\right)^x \) describes an exponential decay because the base \( \frac{1}{2} \) is between 0 and 1. On the other hand, \( g(x) = 2^x \) represents exponential growth because the base, 2, is greater than 1.
These functions are crucial in modeling real-world phenomena, including population growth and radioactive decay. Understanding how these structures behave sets the foundation for more complex mathematical concepts. Each exponential function has a horizontal asymptote, typically the x-axis, which the graph approaches but never truly touches or crosses.
By visualizing them on a graph, you get a clear picture of how unbelievably fast they can grow or shrink based on their respective bases.

Graphing Techniques

Graphing techniques help us visualize and interpret functions, providing insight that equations alone cannot. With functions like \( f(x) = (\frac{1}{2})^x \) and \( g(x) = 2^x \), graphing becomes an indispensable tool. To begin plotting these functions, it is crucial to select an appropriate scale and view, such as the specified window -

• \(-4 \leq x \leq 4\)
• \(-1 \leq y \leq 8\)

This ensures the complete behaviors of the functions are visible. When plotting, it's important to identify key characteristics. For exponential growth, you will notice steeply rising curves, whereas exponential decay yields curves that rapidly approach but do not hit the horizontal axis.
Technological tools such as graphing calculators or online graph tools simplify this process, allowing immediate access to function visualization without manual plotting. They let you manipulate the view to closely analyze specific portions of the graph, facilitating deeper understanding.
Graphing also provides a visual depiction of symmetry or other relationships, much like discovering how \( f(x) \) and \( g(x) \) become mirror images of each other.

Symmetry about the y-axis

Symmetry about the y-axis is a powerful concept that tells us a lot about a function's graphical behavior. If you imagine the y-axis as a vertical mirror, symmetry about this axis means one side of the graph is a mirror image of the other. In the context of our functions,

• \( f(x) = (\frac{1}{2})^x \)
• \( g(x) = 2^x \)

we observe that they are perfectly symmetrical to each other when reflected across the y-axis.
Mathematically, this symmetry indicates that for every point \((x, y)\) on \( g(x) \), there is a corresponding point \((-x, y)\) on \( f(x) \). This aspect tells us they are almost identical but flipped horizontally. Moreover, understanding this aspect means recognizing that \( g(x) \) and \( f(x) \) or any equivalent \( k(x) = 2^{-x} \) are reflections of each other.
This symmetry is an inviting property in solving equations and generating new functions because it hints at equivalences and transformations that might not be immediately obvious. It's an exciting characteristic that adds depth and insight into the graphical relationship between these exponential functions.