Problem 53 - Algebra and Trigonometry

Question

In Exercises \(53-58,\) begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$ g(x)=\log _{2}(x+1) $$

Step-by-Step Solution

Answer

The vertical asymptote of the given function is x=-1. The domain of the function is \((-1,\infty)\) and the range is \((-\infty,\infty)\).

Step 1: Graph the Basic Function

Start by graphing the basic function \(f(x)=\log_{2}x\). The graph passes through the point (1,0), goes to infinity as x goes to infinity, and approaches negative infinity as x approaches 0 from the right. The line x=0 (y-axis) is a vertical asymptote for this basic function.

Step 2: Identify the Transformation

The given function \(g(x)=\log_{2}(x+1)\) is a horizontal shift of the basic logarithmic function. The term \((x+1)\) indicates that the graph of \(f(x)\) is shifted one unit to the left.

Step 3: Graph the Transformed Function

Sketch the graph of the transformed function \(g(x)=\log_{2}(x+1)\) based on the basic function \(f(x)=\log_{2}x\). The graph of \(g(x)\) passes through the point (0,0), goes to infinity as x goes to infinity, and approaches negative infinity as x approaches -1 from the right.

Step 4: Determine the Vertical Asymptote

The line x=-1 is a vertical asymptote for the transformed function \(g(x)=\log_{2}(x+1)\), because the function approaches negative infinity as x approaches -1.

Step 5: Determine the Domain and Range

The domain of the function \(g(x)=\log_{2}(x+1)\) corresponds to all real numbers x where \(x+1 > 0\), or \(x > -1\). Therefore, the domain is \((-1,\infty)\). The range of a logarithmic function is always \((-\infty,\infty)\), which is the same for the transformed function \(g(x)=\log_{2}(x+1)\).

Key Concepts

Function Transformations

Understanding function transformations is key to graphing functions with ease. For logarithmic functions, transformations can include translations, stretches, and reflections, similar to other function types. Let’s break down these changes using our example function,

• Translation: The logarithmic function transformation here involves a horizontal shift. The given function is \( g(x) = \log_{2}(x+1) \). The \(+1\) inside the parentheses indicates a horizontal shift to the left by 1 unit.
• No Vertical Change: Since there is no coefficient affecting the \(g(x)\) or any addition/subtraction outside the logarithm, the vertical position of the graph remains unchanged.

Understanding these transformations helps in plotting the function correctly in any context. Graphing logarithmic functions becomes much easier when you recognize these modifications and apply them step by step.

Vertical Asymptote

A vertical asymptote is a critical feature of logarithmic functions and is where the function tends to infinity. For the function \( f(x) = \log_{2}x \), the vertical asymptote is the line \( x = 0 \) because the log function approaches negative infinity as \( x \) approaches zero from the right.
For the transformed function \( g(x) = \log_{2}(x+1) \), the addition inside the log argument shifts the asymptote left by 1 unit. This gives our new vertical asymptote as \( x = -1 \).
Knowing the location of the vertical asymptote helps us understand the behavior of the function as it approaches certain x-values.

Domain and Range

The domain and range of a function tell us where the function exists and how its outputs behave. It's crucial to identify these to fully understand any function.
For \( g(x) = \log_{2}(x+1) \), the domain consists of all x-values that make the argument of the logarithm positive. This means \( x+1 > 0 \), leading to \( x > -1 \). So, the domain of \( g(x) \) is \((-1, \infty)\).

• Domain: \((-1, \infty)\)
• Range: Logarithmic functions typically have a range of \((-\infty, \infty)\), and this range remains the same for the transformed function.

Understanding these aspects ensures that the function is used correctly over its possible real-number inputs and outputs.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and have a variety of applications in real-world situations like measuring the intensity of sound (decibels) or the magnitude of earthquakes (Richter scale). The basic form \( f(x) = \log_{b}(x) \) has certain characteristics:

• Passes Through: The point (1,0) because any number raised to zero equals 1.
• Vertical Asymptote: at \( x = 0 \). This is due to the fact that a logarithm function is undefined for non-positive values.
• Behavior: The graph continues indefinitely as \( x \to +\infty \), growing without bound.

The above features apply to any base \(b>1\). It’s these unique properties that set logarithmic functions apart and allow them to model precise real phenomena.