Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. Then sketch its graph.\(f(x)=\ln (3-x)\)

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \ln(3-x)\) is \(x \in (-\infty, 3)\), the vertical asymptote is at \(x = 3\), and the x-intercept is at \(x = 2\).

Step 1: Find the Domain

The domain of a logarithmic function is the set of all real numbers \(x\) where the argument of the logarithm is positive. For the function \(f(x) = \ln(3-x)\), we set the argument \(3-x > 0\) which yields \(x < 3\). So, the domain of \(f(x)\) is \(x \in (-\infty, 3)\).

Step 2: Identify the Vertical Asymptote

The vertical asymptote of a logarithmic function occurs at the boundary of its domain. In this case, according to the domain \(x \in (-\infty, 3)\), the vertical asymptote is the line \(x=3\).

Step 3: Find the x-Intercept

The x-intercept of a function is the x-value where the function equals zero. So, set \(f(x) = 0\) and solve for \(x\). We have \(0 = \ln(3-x)\). Converting this logarithmic equation to an exponential form gives us \(1 = 3-x\), so \(x = 2\). Therefore, the x-intercept is \(x = 2\).

Step 4: Sketch the Graph

Begin by drawing the vertical asymptote at \(x = 3\). Next, mark the x-intercept at \(x = 2\). The basic shape of a decreasing logarithmic function will result in a curve that gets increasingly close to the asymptote as \(x\) approaches 3 from the left but never reaches it. The function approaches negative infinity as \(x\) approaches the asymptote.

Key Concepts

Domain of a Logarithmic Function

Understanding the domain of a logarithmic function is vital for solving equations and graphing. In mathematical terms, the domain refers to the permissible values that we can plug into the function without producing an undefined result.

In the case of logarithms, this means that the argument of the logarithm—the number inside the log function—must be greater than zero because the logarithm of a non-positive number is undefined. For the function presented, the argument of the logarithm is \(3-x\), so the condition for the argument is \(3-x>0\). To find the values of \(x\) that make this true, you solve the inequality, yielding \(x<3\). Hence, the domain of the function \(f(x) = \ln(3-x)\) is all real numbers less than 3, mathematically expressed as \(x \in (-\infty, 3)\).

To ensure students fully grasp this concept, it's important to clarify that the argument of the logarithmic function can never reach zero, and certainly cannot be negative. This is a fundamental property of logarithmic functions and is key to understanding their behavior.

Vertical Asymptote

The concept of a vertical asymptote is often encountered in logarithmic functions. An asymptote is a line that a curve approaches but never touches or crosses. For the function \(f(x) = \ln(3-x)\), we've established that the domain restricts \(x\) to values less than 3. Thus, as \(x\) gets closer to 3 from the left, the \(\ln(3-x)\) function decreases without bound, getting infinitely large in the negative direction.

Because the value of \(f(x)\) will never equal the line \(x=3\), this line is considered a vertical asymptote. On a graph, you'd draw this as a dashed vertical line at \(x=3\), illustrating a boundary that the function cannot cross. For students visualizing this concept, imagine it as a barrier that the graph eternally approaches but never actually reaches, symbolizing the limit of the function's domain.

X-Intercept

The \(x\)-intercept of a function is the point where the graph of the function crosses the \(x\)-axis. To find this point for a logarithmic function like \(f(x) = \ln(3-x)\), you set \(f(x)\) to zero and solve for \(x\). The equation \(0 = \ln(3-x)\) tells us that \(e^0 = 3-x\), because the inverse of the natural logarithm is the exponential function with base \(e\). Given that \(e^0 = 1\), you easily find that \(x = 2\).

The \(x\)-intercept is a crucial point in sketching the graph because it provides a reference point from which the curve extends. Notably, a logarithmic function can have at most one \(x\)-intercept, as the nature of these functions allows them to cross the \(x\)-axis just once. It's a simple yet essential aspect of the function's graph and is key in graphing the function correctly.

Graphing Logarithmic Functions

Graphing logarithmic functions, such as \(f(x) = \ln(3-x)\), involves understanding the function's domain, vertical asymptote, and \(x\)-intercepts. Start your graph with these fundamental components. Draw a dashed vertical line to represent the vertical asymptote at \(x = 3\). Next, plot the \(x\)-intercept by marking a point on the \(x\)-axis at \(x=2\).

Remember that the curve of the logarithmic function will move towards the vertical asymptote as \(x\) approaches 3 and will become increasingly steep. The function will continually decrease to negative infinity without touching the vertical asymptote. By marking several points that satisfy the function and connecting them smoothly, you can draw an accurate graph of the logarithmic function.

A handy tip for students struggling to sketch these graphs is to plot a few easy-to-calculate points. For instance, using \(x=0\), \(x=1\), and so forth can give immediate points through which the function's curve should pass. With practice, drawing graphs of logarithmic functions can become as straightforward as sketching lines.