Problem 41 - College Algebra

Question

Graph \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\log _{1 / 2} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graphs of \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\log _{1 / 2} x\) are inverses of each other. The graph of \(f(x)\) starts at the upper left, passes through (0,1), and approaches the x-axis as it moves to the right. The graph of \(g(x)\) starts at the top right, passes through (1,0), and approaches the y-axis as it moves to the right.

Step 1: Key Features of Exponential Function

Identify the key features of the exponential function \(f(x)=\left(\frac{1}{2}\right)^{x}\). This function has a y-intercept at (0,1). The x-axis is a horizontal asymptote, which the graph approaches but never crosses. As \(x\) increases, \(f(x)\) decreases towards 0. As \(x\) decreases, \(f(x)\) increases towards infinity.

Step 2: Graph the Exponential Function

Sketch the graph of \(f(x)=\left(\frac{1}{2}\right)^{x}\) using its key features. The graph should start from the upper-left corner of the coordinate plane (signifying \(x\) approaching negative infinity and \(f(x)\) approaching infinity), pass through the y-intercept (0,1), and then approach the x-axis as \(x\) tends towards positive infinity indicating \(f(x)\) is approaching zero.

Step 3: Key Features of Logarithmic Function

Identify the key features of the logarithmic function \(g(x)=\log _{1 / 2} x\). This function has an x-intercept at (1,0). The y-axis acts as a vertical asymptote, which the graph approaches but never crosses. As \(x\) increases, \(g(x)\) decreases towards negative infinity. As \(x\) decreases towards 0, \(g(x)\) increases towards positive infinity.

Step 4: Graph the Logarithmic Function

Sketch the graph of \(g(x)=\log _{1 / 2} x\) using its key features. The graph should start from the upper-right corner of the coordinate plane (signifying \(x\) approaching 0 and \(g(x)\) approaching positive infinity), pass through the x-intercept (1,0), and then approach the y-axis as \(x\) tends towards positive infinity indicating \(g(x)\) is approaching negative infinity.

Key Concepts

Exponential Functions

An exponential function is a mathematical expression where the variable is in the exponent. In the function \( f(x)=\left(\frac{1}{2}\right)^{x} \), some important characteristics help us understand its behavior. First, it is important to note the base, \( \frac{1}{2} \), is a fraction between 0 and 1, which means the function will be decreasing.
This function has a y-intercept at the point (0,1). The point (0,1) is where the graph crosses the y-axis, meaning when \( x = 0 \), \( f(x) \) equals 1.
Additionally, the graph of this function has a horizontal asymptote at the x-axis. This horizontal line signifies that as \( x \) becomes very large in the positive direction, \( f(x) \) approaches 0. However, it will never actually touch or cross the x-axis.

• For \( x \) increasing, \( f(x) \) decreases towards 0.
• For \( x \) decreasing, \( f(x) \) grows towards infinity.

Understanding these features allows us to sketch the graph accurately, recognizing its behavior across different quadrants.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. In our example, \( g(x)=\log_{1/2}x \), the base is \( \frac{1}{2} \). It's crucial to understand that the base being between 0 and 1 causes the function to decrease as \( x \) increases.
Important features of this logarithmic function include an x-intercept at (1,0). This point indicates that when \( g(x) \) is 0, the value of \( x \) is 1. This is the same point where an exponential function of the same base intersects the y-axis.
The y-axis serves as a vertical asymptote, meaning as \( x \) approaches 0, \( g(x) \) shoots towards positive infinity. In simpler terms, the graph moves upwards steeply but never actually meets or crosses the y-axis.

• For increasing \( x \), the function tends toward negative infinity.
• For decreasing \( x \), but approaching zero, it ascends toward positive infinity.

By identifying these features, we can understand the graph’s structure, enabling us to correctly sketch and interpret the overall behavior of any logarithmic curve.

Rectangular Coordinate System

The rectangular coordinate system is a fundamental concept in graphing functions, consisting of a two-dimensional plane formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). This system allows us to plot points using ordered pairs \(x, y\), where \(x\) indicates the horizontal position and \(y\) the vertical position.
In the context of this exercise, the exponential and logarithmic functions are plotted together in this system. The features and behaviors identified in these functions—such as asymptotes, intercepts, and the direction of the graphs—are interpreted visually within the coordinate system.
By using this system, multiple functions can be graphed in the same space, helping identify intersections and compare behaviors.

• Intercepts are points where the graph crosses an axis.
• Asymptotes indicate lines that the graph approaches but never touches.
• The graphing system allows for visual comparison and analysis of different functions.

Mastering the rectangular coordinate system is essential for interpreting and understanding how different types of functions behave and interact graphically.