Problem 63 - College Algebra with Modeling and Visualization

Question

Sketch a graph of \(f\) $$f(x)=\log _{2} x$$

Step-by-Step Solution

Answer

Sketch shows a curve passing through (1,0), increasing through (2,1), and has a vertical asymptote at x=0.

Step 1: Identify the type of function

The given function is a logarithmic function, which is written as \( f(x) = \log_2 x \). This means it is the logarithm of \( x \) with base 2.

Step 2: Determine the domain

Logarithmic functions are only defined for positive values of \( x \). Thus, the domain of \( f(x) = \log_2 x \) is \( x > 0 \).

Step 3: Identify key points

To sketch the graph, it's helpful to identify key points of the function. For \( f(x) = \log_2 x \):1. When \( x = 1 \), \( f(x) = \log_2 1 = 0 \).2. When \( x = 2 \), \( f(x) = \log_2 2 = 1 \).3. When \( x = \frac{1}{2} \), \( f(x) = \log_2 \frac{1}{2} = -1 \).

Step 4: Determine the behavior as x approaches boundaries

As \( x \) approaches 0, \( f(x) = \log_2 x \) decreases without bound, meaning \( f(x) \to -\infty \) as \( x \to 0^+ \). As \( x \to \infty \), \( f(x) \to \infty \). This indicates a vertical asymptote at \( x = 0 \) and an increasing logarithmic growth for larger values of \( x \).

Step 5: Sketch the graph

Using the key points and known behavior, sketch the curve. Start at the point \( (1,0) \), which is on the x-axis, showing where the graph crosses it. Draw the graph smoothly increasing through \( (2,1) \) and continuing upwards, while approaching negative infinity as it nears the y-axis (vertical asymptote at \( x=0 \)). The graph should show a slow, continuous increase without crossing or touching the y-axis.

Key Concepts

Graph Sketching

Sketching the graph of a logarithmic function, like \( f(x) = \log_2 x \), can help visualize how the function behaves across different values of \( x \). A logarithmic curve has a distinct shape and characteristics.
To begin sketching, identify critical points that help plot the curve. Start by noting where the curve meets the x-axis. In this case, at point \((1,0)\). This is because \(\log_2 1 = 0\).

• Next, include some other key points, such as \((2,1)\) because \(\log_2 2 = 1\).
• You can also consider \((\frac{1}{2}, -1)\) since \(\log_2 \frac{1}{2} = -1\).

These points help form a rough outline of the function.
Additionally, understand the curve's behavior near the boundaries of its domain. As \(x\) approaches zero, the function's value \(f(x)\) drops sharply towards negative infinity. Meanwhile, as \(x\) increases, \(f(x)\) grows unbounded, moving towards positive infinity.
With these insights, sketch a smooth curve that:

• Starts from the vertical asymptote at \(x=0\)
• Passes through the noted points
• Extends upwards, indicating the increasing nature of the logarithm

A careful sketch considering these points and behavior will yield a precise visual representation of a logarithmic function.

Logarithmic Function Properties

Logarithmic functions have unique properties that make them a fundamental tool in mathematics.
The core function \( f(x) = \log_b x \) where \( b \) is the base of the logarithm, has several key characteristics:

• The domain of logarithmic functions is strictly for positive real numbers, \( x > 0 \).
• These functions have a range of all real numbers, meaning the output can be any real number depending on the input.

Furthermore, logarithmic functions exhibit a vertical asymptote at \( x = 0 \). As \( x \) gets closer to zero, the logarithmic value becomes increasingly negative, moving towards \(-\infty\).
Another critical property is their increasing nature. As \( x \) increases, \( \log_b x \) also increases without bound.
For even minor changes in \( x \), the value of \( f(x) \) starts to grow. This logarithmic growth implies a slow rate of increase when compared to exponential functions but shows consistent upward movement.
Remember that logarithms can only be calculated for positive numbers. Negative or zero values are not undefined, giving logarithms a unique behavior in comparison to other mathematical functions.

Domain and Range Analysis

Understanding the domain and range of a function is crucial for sketching its graph and predicting its behavior.
For the logarithmic function \( f(x) = \log_2 x \), the domain consists of all positive numbers: \( x > 0 \). This is because logarithms of negative numbers or zero are undefined in real number systems.
The practical implication of this domain is that on the graph, you will not see the curve going left of the y-axis.
The range of a logarithmic function, however, is all real numbers. This implies that the output values for \( f(x) \) can cover every real number from negative to positive infinity. On a graph, the curve extends upwards infinitely as \( x \) increases, covering more and more of the y-axis.

• In essence, while the domain restricts the input values, the range tells us that the growth potential of logarithms is unlimited in terms of output.
• These characteristics define the way the graph climbs upwards without restrictions, always starting from a very low point near the vertical asymptote.

With both domain and range, the limits and potentials of the function are well-defined, allowing for accurate graph sketching and deeper understanding of logarithmic behavior.