Problem 60 - Calculus

Question

If \(g(x)=\ln (a x+2),\) where \(a \neq 0,\) what is the effect of increasing \(a\) on the vertical asymptote?

Step-by-Step Solution

Answer

Increasing \( a \) shifts the vertical asymptote closer to zero.

Step 1: Understanding the Function

Given the function \( g(x) = \ln(ax + 2) \), we know the natural logarithm function, \( \ln(x) \), has a domain for \( x > 0 \). Therefore, the expression inside the logarithm, \( ax + 2 \), must be positive. This leads us to the inequality \( ax + 2 > 0 \).

Step 2: Solving the Inequality

From the inequality \( ax + 2 > 0 \), we solve for \( x \) to determine the domain of the function. This gives \( ax > -2 \) or \( x > -\frac{2}{a} \). This indicates the vertical asymptote of \( g(x) \) is at \( x = -\frac{2}{a} \).

Step 3: Assessing the Effect of Increasing \( a \)

When we increase the value of \( a \), the position of the vertical asymptote \( x = -\frac{2}{a} \) is affected. As \( a \) increases, since \( a eq 0 \), the magnitude of \( -\frac{2}{a} \) decreases, moving the vertical asymptote closer to zero.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental mathematical function often used in calculus and algebra. It is the inverse operation of exponentiating the base \( e \) (approximately 2.718), which is also known as Euler's number. In simpler terms, the natural logarithm tells us the power to which \( e \) must be raised to result in a given number.

Here are some key features of the natural logarithm function:

• Defined only for positive numbers \( x > 0 \).
• Passes through the point \( (1, 0) \) because \( \ln(1) = 0 \).
• Increases slowly and never drops below the x-axis, but goes to infinity as \( x \) increases.

Understanding this function helps in grasping the behavior of functions that incorporate it, such as \( g(x) = \ln(ax + 2) \), where certain transformations can shift or stretch the graph.

Domain of the Function

The domain of a function is crucial because it defines the set of input values for which the function is mathematically valid. For functions involving the natural logarithm, like \( g(x) = \ln(ax + 2) \), the domain is determined by the condition that the argument of the logarithm, here \( ax + 2 \), must be greater than 0.

Following the inequality \( ax + 2 > 0 \), solving for \( x \) gives us \( x > -\frac{2}{a} \). Thus, the domain of the function is all real numbers \( x \) such that \( x \) is greater than \( -\frac{2}{a} \).

The line \( x = -\frac{2}{a} \) is where the input to the logarithm switches from negative to positive, which corresponds to where a vertical asymptote occurs. This boundary indicates the point beyond which the function is defined and helps in visualizing or graphically representing the function.

Effect of Parameter Change

In the context of \( g(x) = \ln(ax + 2) \), changing the parameter \( a \) has a specific effect on the function, particularly on the position of its vertical asymptote. A vertical asymptote is a line \( x = c \) where a function approaches but never actually reaches \( c \), often leading to values that grow infinitely large or small.

Here’s how adjusting \( a \) affects the function:

• Increasing \( a \) moves the vertical asymptote, which is initially at \( x = -\frac{2}{a} \), closer to zero.
• As \( a \) gets larger, \( -\frac{2}{a} \) becomes a smaller negative number, shrinking the domain from the left.
• This means the function starts being defined at a point that is less negative.

Therefore, understanding the effect of parameter changes is essential for predicting and explaining the behavior of functions geometrically and algebraically.