Problem 15 - Calculus for Biology and Medicine

Question

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=\ln (x+1) $$

Step-by-Step Solution

Answer

The graph of \( y = \ln(x+1) \) is a logarithmic curve with a vertical asymptote at \( x = -1 \), starting from the x-axis and increasing to infinity as \( x \to \infty \).

Step 1: Understand the Function

The function given is \( y = \ln(x+1) \), where \( \ln \) denotes the natural logarithm. The natural logarithm function \( \ln(x) \) is defined only for \( x > 0 \). Therefore, \( x+1 > 0 \) defines the domain for our function.

Step 2: Determine the Domain

For the function \( y = \ln(x+1) \) to be valid, \( x+1 \) must be greater than zero, which simplifies to \( x > -1 \). Hence, the domain of the function is \( x > -1 \).

Step 3: Identify Key Points

Calculate a few key points for sketching: - At \( x = 0 \), \( y = \ln(0+1) = \ln(1) = 0 \).- At \( x = 1 \), \( y = \ln(1+1) = \ln(2) \approx 0.693 \).- As \( x \to \infty \), \( y \to \infty \).- As \( x \to -1^+ \), \( y \to -\infty \).

Step 4: Asymptotic Behavior

Note the behavior of the function as \( x \) approaches any vertical asymptotes. Since for \( x = -1 \), \( y \to -\infty \), there is a vertical asymptote at \( x = -1 \).

Step 5: Sketch the Graph

Begin by drawing the vertical asymptote at \( x = -1 \). Plot the key points we calculated earlier. Starting from the point \( (0, 0) \), increase towards the right, moving towards \( (1, \ln(2)) \) and eventually approaching infinity as \( x \) increases, following the log curve shape. Approach the vertical asymptote from the right as \( x \to -1^+ \), showing the graph decreases sharply towards \(-\infty\).

Step 6: Complete the Analysis

Your graph should reflect all key characteristics: a vertical asymptote at \( x = -1 \), intercept at the point (0, 0), and the general shape of a logarithmic curve that increases and moves towards the right forever.

Key Concepts

Domain of a Function

When talking about the domain of a function, we're referring to all the possible values of \( x \) that can go into a function to produce a valid output, \( y \). For our function \( y = \ln(x+1) \), we need to ensure that the expression inside the logarithm, \( x+1 \), is positive. This is because the natural logarithm \( \ln(x) \) is undefined for values of \( x \leq 0 \).
In this case, the condition \( x+1 > 0 \) simplifies to \( x > -1 \). Therefore, the function's domain includes all real numbers greater than \(-1\), not including \(-1\) itself. To represent this in interval notation, we write the domain as \((-1, \infty)\).
This simply means you can plug in any number bigger than \(-1\) into the function to get a valid output. Understanding the domain helps us avoid inputs that would result in undefined behavior for our function.

Vertical Asymptote

A vertical asymptote is a vertical line that a graph of a function approaches but never touches or crosses. These are essential in graphing as they indicate points where the function doesn't exist. For the function \( y = \ln(x+1) \), the vertical asymptote is where the natural logarithm reaches negative infinity, signifying a point just before the function becomes undefined.

For the natural logarithm function based at \( (x+1) \), this occurs as \( x \) approaches \(-1\) from the right (i.e., \( x \to -1^+ \)). As \( x \) gets closer to \(-1\), the value of \( y \) decreases without bound, moving towards negative infinity. This makes the line \( x = -1 \) a vertical asymptote.

• Vertical asymptotes can tell us about the behavior of functions near critical points.
• They are not part of the domain and are typically represented as dashed or dotted lines on graphs.
• For \( y = \ln(x+1) \), \( x = -1 \) defines the boundary where the function ceases to exist.

Being mindful of asymptotes is crucial when sketching graphs to accurately represent where a function sharply drops or shoots up.

Key Points on a Graph

Key points on a graph provide us with essential information that helps sketch the curve accurately. For the logarithmic function \( y = \ln(x+1) \), identifying these points is crucial to represent the curve's shape and direction.

Let's start by examining a few foundational points:

• At \( x = 0 \), \( y = \ln(1) = 0 \). This point \((0,0)\) is crucial as it marks where the graph crosses the y-axis.
• At \( x = 1 \), \( y = \ln(2) \approx 0.693 \). This point is slightly above zero, indicating an upward, increasing trend.
• For large values of \( x \), as \( x \to \infty \), \( y \to \infty \). This by implies that the curve stretches indefinitely towards the right along the x-axis.
• Conversely, as \( x \to -1^+ \), \( y \to -\infty \). The graph approaches this sharp decline towards negative infinity near our vertical asymptote at \( x=-1 \), showcasing its asymptotic behavior.

With these key points and patterns, each provides a checkpoint-for-charting the graph. It allows you to establish the overall direction and behavior of the function, predicting its long-term trends and any critical points along the axes.