Problem 52 - Intermediate Algebra

Question

Graph \(f(x)=x, f(x)=(0.5)^{x}\), and \(f(x)=\log _{0.5} x\) on the same set of axes.

Step-by-Step Solution

Answer

Graph the linear, exponential, and logarithmic functions, noting that the exponential decays, the logarithm is its inverse, and both cross at key points like (1,1) and (2,-2).

Step 1: Identify Functions

The functions to graph are: \(f(x) = x\), which is a linear function; \(f(x) = (0.5)^x\), which is an exponential decay function; and \(f(x) = \log_{0.5}x\), which is a logarithmic function with base 0.5.

Step 2: Determine Key Points

For \(f(x) = x\), the key points are any points on the line where the function and input are equal, such as \((0,0), (1,1), (-1,-1)\). For \(f(x) = (0.5)^x\), key points include \((0,1), (1,0.5), (-1,2)\). For \(f(x) = \log_{0.5}x\), key points might be \((1,0), (0.5,1), (2,-1)\).

Step 3: Graph the Linear Function

The graph of \(f(x) = x\) is a straight line passing through the origin with a slope of 1. Plot it by connecting these key points: \((0,0), (1,1), (-1,-1)\).

Step 4: Graph the Exponential Function

The graph of \(f(x) = (0.5)^x\) shows exponential decay. Plot the points \((0, 1)\), \((1, 0.5)\), and \((-1, 2)\). Connect these points with a smooth curve that approaches the x-axis but never touches it.

Step 5: Graph the Logarithmic Function

The graph of \(f(x) = \log_{0.5}x\) is the inverse of the exponential decay function. Plot the points \((1, 0)\), \((0.5, 1)\), and \((2, -1)\). The curve approaches the y-axis but does not touch it and decreases as x increases, reflecting its negative slope for x > 1.

Step 6: Analyze the Intersection Points

Examine where the functions cross each other or the axes. \(f(x) = x\) intersects \( f(x) = (0.5)^x\) and \(f(x) = \log_{0.5} x\) at their respective key points such as \((1,1)\) and \((2,-2)\), noting the zero points and behavior of the curves.

Key Concepts

Understanding Linear Functions

A linear function is one of the simplest types of functions you can encounter in algebra. It is represented by the equation \( f(x) = ax + b \), where \( a \) and \( b \) are constants. The function is called "linear" because its graph is a straight line. For the specific case of \( f(x) = x \), which is the simplest linear function possible, we have a slope \( a = 1 \) and an intercept \( b = 0 \). This means:

• The graph passes through the origin, or point \((0,0)\).
• The slope, which is the steepness or angle of the line, is 1. This means for every unit increase in \( x \), \( y \) increases by 1 unit too.

To graph \( f(x) = x \), you simply plot key points such as \((0,0)\), \((1,1)\), and \((-1,-1)\), and draw a straight line through them. This line will extend infinitely in both directions, maintaining its slope of 1. Linear functions are often used to model relationships with constant rates of change, making them crucial in understanding the basics of function graphing.

Exploring Exponential Decay Functions

Exponential decay functions describe processes that decrease or decay at a rate proportional to their current value. These functions have the general form \( f(x) = a(0 < b < 1)^x \). For the problem at hand, the function \( f(x) = (0.5)^x \) gives us an example of how quantities decrease over time. Here's what you need to know:

• The base \( b = 0.5 \) tells you that the function’s value halves as \( x \) increases by 1.
• It crosses the y-axis at the point \( (0,1) \), because any number to the power of 0 is 1.
• The graph approaches the x-axis but never quite touches it, showing that while values get very small, they never reach zero.

To sketch the graph, use points like \((0,1)\), \((1,0.5)\), \((-1,2)\), connecting them with a smooth curve. This curve will demonstrate the exponential decrease, or "decay," moving closer to the x-axis without intersecting it. Exponential decay functions are often used in contexts like radioactive decay and population studies.

Understanding Logarithmic Functions

Logarithmic functions serve as inverses of exponential functions. For a logarithmic function \( f(x) = \log_b x \) with base \( b \), it answers the question: "To what power do we raise \( b \) to get \( x \)?" Specifically, the function \( f(x) = \log_{0.5} x \) is considered here. Historically, logarithms have transformed how we handle multiplication and division, enabling calculations with large numbers.For \( f(x) = \log_{0.5} x \):

• It crosses the x-axis at \((1,0)\), because any logarithmic function equals zero when its input is 1.
• As the input doubles, the output decreases negatively. This is because \(0.5^x\) grows smaller as \(x\) increases, translating to negative outputs in the logarithm.
• The curve never touches the y-axis, reflecting the area where logarithms aren’t defined for non-positive numbers.

The graph is characterized by its decreasing slope and points such as \((1,0)\), \((0.5,1)\), and \((2,-1)\), illustrating its decreasing behavior. Logarithmic functions like this one help in understanding scales such as the Richter scale, which measures earthquake magnitudes.