Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, state the domain and the vertical asymptote of the function. \(g(x)=-\ln (3 x+9)-7\)

Step-by-Step Solution

Answer

Domain: \(x > -3\); Vertical Asymptote: \(x = -3\).

Step 1: Identify the Argument of the Logarithm

The given function is \( g(x) = -\ln(3x + 9) - 7 \). We need to start by identifying the argument inside the logarithmic function, which is \(3x + 9\).

Step 2: Determine the Domain of the Function

For a logarithmic function \( \ln(a) \), the argument \( a \) must be greater than zero. Therefore, set the inequality \( 3x + 9 > 0 \). Solve this inequality: \( 3x > -9 \), and then divide by 3 to find \( x > -3 \). Hence, the domain of the function is \( x > -3 \).

Step 3: Identify the Vertical Asymptote

The vertical asymptote of a logarithmic function occurs where the argument equals zero. Set \( 3x + 9 = 0 \). Solving this equation gives \( 3x = -9 \), and \( x = -3 \). Hence, the vertical asymptote is at \( x = -3 \).

Step 4: Write the Domain and Vertical Asymptote

Summarize the findings: The domain of the function is \( x > -3 \), and the vertical asymptote is at \( x = -3 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are a special type of mathematical functions that are the inverse of exponential functions. They are often written in the form \( \log_b(x) \), where \( b \) is the base of the logarithm. Common bases include 10 (common logarithm), \( e \) (natural logarithm), and 2 (binary logarithm). In this exercise, we encounter the natural logarithm \( \ln(x) \).

One important property of logarithmic functions is that they are only defined for positive values of \( x \), meaning the argument of the function must be greater than zero. This implies that the range of a logarithmic function is all real numbers, but the domain is strictly positive values. This characteristic significantly influences how we determine the properties of the function, such as the vertical asymptotes and the domain of the function.

Inequalities

Inequalities are mathematical expressions used to show the relative size of two values. They are crucial in determining the domain of logarithmic functions. In the given function, \( g(x) = -\ln(3x + 9) - 7 \), we focus on the inequality \( 3x + 9 > 0 \) to find the domain.

Solving inequalities may involve moving terms across the equation or dividing and multiplying to isolate variables, much like solving equations. Here, we move 9 to the other side, giving us \( 3x > -9 \). Following that, we divide both sides by 3 to arrive at \( x > -3 \).

This inequality solution tells us exactly which values of \( x \) will render the logarithmic function valid by ensuring its argument is positive. Understanding how to manipulate and solve inequalities is essential for analyzing and understanding various functions.

Vertical Asymptote

A vertical asymptote in a function is a line \( x = c \) where the function tends to infinity, creating a gap or division in the graph. In the case of logarithmic functions, a vertical asymptote occurs where the argument of the logarithm is zero. For our function \( g(x) = -\ln(3x + 9) - 7 \), setting the argument \( 3x + 9 \) equal to zero finds this asymptote.

Solving \( 3x + 9 = 0 \) involves subtracting 9 from both sides, leaving us with \( 3x = -9 \). Dividing by 3, we get \( x = -3 \). Hence, the vertical asymptote is located at \( x = -3 \).

Vertical asymptotes are significant because they mark the boundary of the function's domain and imply that the function will approach infinity or negative infinity as \( x \) nears this point. They are key features in the graph of logarithmic and rational functions.

Function Domain

The domain of a function constitutes all potential \( x \)-values for which the function is defined. For logarithmic functions like \( g(x) = -\ln(3x + 9) - 7 \), the domain is determined by ensuring the argument of the logarithm is positive. In this instance, we solve the inequality \( 3x + 9 > 0 \), which leads to \( x > -3 \).

The domain can also be expressed in interval notation, where for this function, it would be \((-3, \infty)\). This indicates that \( x \) can take any value greater than \(-3\), but not \(-3\) itself.

Understanding the domain is vital in developing a deeper comprehension of function behavior. It tells us for which values a function exists and is valid, setting the stage for further analysis and graphing of the function under consideration.