Problem 69 - Algebra for College Students

Question

Graph \(f(x)=x, f(x)=2^{x}\), and \(f(x)=\log _{2} x\) on the same set of axes.

Step-by-Step Solution

Answer

Graph the functions on the same axes, interpreting them as linear, exponential, and logarithmic behaviors.

Step 1: Understand Each Function

First, identify the three functions to be graphed. The function \(f(x) = x\) is a linear function, \(f(x) = 2^x\) is an exponential function, and \(f(x) = \log_2 x\) is a logarithmic function.

Step 2: Choose a Set of Axes

Since all functions involve base 2, a reasonable choice of axes could be \(-2 \leq x \leq 4\) and \(-4 \leq y \leq 16\) to capture the behavior of these functions from negative to positive values of \(x\).

Step 3: Plot the Linear Function \(f(x) = x\)

Begin by plotting the linear function \(f(x) = x\), which is a straight line passing through the origin with a slope of 1. Points like \((-2, -2)\), \((0, 0)\), and \((4, 4)\) can help plot the line.

Step 4: Plot the Exponential Function \(f(x) = 2^x\)

Next, plot the exponential function \(f(x) = 2^x\). For example, \(2^{-2} = 0.25\), \(2^0 = 1\), \(2^2 = 4\), and \(2^4 = 16\). This function increases rapidly as \(x\) increases.

Step 5: Plot the Logarithmic Function \(f(x) = \log_2 x\)

Now, plot the logarithmic function \(f(x) = \log_2 x\). Note that \(\log_2 1 = 0\), \(\log_2 2 = 1\), and \(\log_2 4 = 2\). This function is undefined for \(x \leq 0\), and it increases slowly for positive \(x\).

Step 6: Combine All Graphs

Draw all graphs on the same set of axes. Ensure to label each graph clearly. \(f(x) = x\) will be a straight line, \(f(x) = 2^x\) will rise steeply as \(x\) increases, and \(f(x) = \log_2 x\) will curve upwards slowly for \(x > 0\).

Key Concepts

Linear Functions

Linear functions are one of the most straightforward types of functions you will encounter. A linear function is represented by the equation \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this case, however, the linear function \( f(x) = x \) is what we'll explore. This function has a slope of 1 and a y-intercept of 0, making it pass through the origin.

Key Characteristics:

• The slope of 1 implies that for every unit increase in \( x \), \( f(x) \) also increases by 1 unit. This consistent rate of change makes it easy to plot: for example, point \(( -2, -2)\), through \( (0, 0) \), to \( (4, 4) \).
• Graphically, it is represented by a straight diagonal line. It is symmetrical about the origin because of its slope, which mirrors the x-axis on either side.
• Linear functions do not have maximum or minimum values, nor do they curve or steepen further, making them unique in their predictability.

This simplicity is why linear functions are excellent for modeling relationships with constant rates.

Exponential Functions

Exponential functions may seem more complex at first but they have distinct properties. The general form is \( f(x) = a^x \), where \( a \) is a positive constant. Here, we focus on \( f(x) = 2^x \), indicating that the growth factor is 2.

Important Traits:

• One of their defining features is their rapid growth rate. For instance, starting from a fraction like \( 2^{-2} = 0.25 \), they quickly escalate to \( 16 \) at \( x = 4 \). Each step to the right on the x-axis means the value of \( f(x) \) doubles.
• Exponential functions always have a horizontal asymptote at the x-axis, meaning the graph gets closer and closer to the x-axis as \( x \) becomes negative but never actually touches it.
• They only cross the y-axis at one point, which is the y-intercept, occurring at \( f(0) = 1 \) for \( 2^x \).

In real life, exponential functions model things that grow or decay at accelerating rates, such as populations, investments, or radioactive decay.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they have the form \( f(x) = \log_a x \), where \( a \) is the base of the logarithm. In this context, we examine \( f(x) = \log_2 x \). Understanding logarithmic graphs starts by translating the exponential behavior in reverse.

Key Elements:

• The graph of a logarithmic function passes through the point \( (1, 0)\) because \( a^0 = 1 \), making \( \log_a 1 = 0 \).
• For \( x = 2 \), \( \log_2 2 = 1 \) and similarly, \( \log_2 4 = 2 \), showing how the function increases, albeit slowly, as \( x \) grows.
• Unlike exponential functions, logarithmic ones are undefined for \( x \leq 0 \). Thus, they only exist in the positive quadrant of the coordinate plane.

These functions are practical in understanding phenomena where one quantity scales multiplicatively, such as computing time complexity in algorithms or measuring sound intensity using decibels.