Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Graph each function. $$f(x)=\log _{10}(3-x)$$

Step-by-Step Solution

Answer

The function has a domain of \((-\infty, 3)\), a vertical asymptote at \(x = 3\), and passes through \((0, \log_{10}(3))\).

Step 1: Identify the Domain

The domain of the function is the set of all x-values for which the function is defined. For the function \( f(x) = \log_{10}(3-x) \), the logarithm is only defined for positive arguments. Therefore, we need \( 3-x > 0 \), which simplifies to \( x < 3 \). Thus, the domain of the function is \( (-\infty, 3) \).

Step 2: Identify Vertical Asymptote

The vertical asymptote occurs where the argument of the logarithm is zero because the logarithm is undefined at zero. Therefore, set \( 3-x = 0 \), which gives \( x = 3 \). Hence, there is a vertical asymptote at \( x = 3 \).

Step 3: Find Intercept

To find the y-intercept, evaluate the function at \( x = 0 \). \[ f(0) = \log_{10}(3-0) = \log_{10}(3) \]. This gives the y-intercept as \( (0, \log_{10}(3)) \).

Step 4: Analyze Behavior Near Asymptote

As \( x \) approaches 3 from the left (\( x \to 3^{-} \)), the argument of the log, \( 3-x \), approaches zero. The function \( \log_{10}(3-x) \) will go to \(-\infty\) as \( x \to 3^{-} \).

Step 5: Plot a few key points and draw the graph

Use the information from previous steps to graph the function. Key points include the y-intercept at \( (0, \log_{10}(3)) \). As \( x \to 3^{-} \), the function approaches \(-\infty\), and as \( x \to -\infty \), the function approaches \( \log_{10}(3) \). Connect these points smoothly and ensure the curve becomes steep as it approaches the vertical asymptote at \( x = 3 \).

Key Concepts

Domain of a function

The domain of a function refers to the complete set of possible input values (or x-values) that will produce a valid output from a mathematical function. For logarithmic functions like \( f(x) = \log_{10}(3-x) \), it is crucial to identify which x-values will allow the logarithmic function to be defined.

In this specific case, the expression inside the logarithm \( 3-x \) must be positive because the logarithm of zero or a negative number is undefined in the realm of real numbers. To find when \( 3-x \) is greater than zero, we set up the inequality:

• \( 3 - x > 0 \)
• Solving for \( x \) gives \( x < 3 \)

This reveals that the domain of \( f(x) = \log_{10}(3-x) \) consists of all real numbers less than 3. Therefore, the domain is expressed in interval notation as \( (-\infty, 3) \).

Vertical Asymptote

A vertical asymptote in the graph of a function is a line that the graph approaches but never actually touches. It typically occurs when the function becomes undefined, which for logarithmic functions, happens when their argument is zero.

In our function \( f(x) = \log_{10}(3-x) \), the vertical asymptote is found by setting the argument of the logarithm to zero:

• \( 3 - x = 0 \)
• Solving for \( x \) reveals \( x = 3 \)

This means the vertical asymptote is a vertical line at \( x = 3 \). As the graph approaches this line from the left, it declines rapidly and never crosses the line. This is because the values of the logarithmic function become very large negatively as the function approaches the asymptote.

Y-intercept

The y-intercept of a graph represents the point where the graph of the function intersects the y-axis. This occurs when \( x = 0 \), since the y-axis corresponds to all points where the x-coordinate is zero.

To find the y-intercept for \( f(x) = \log_{10}(3-x) \), we substitute \( x = 0 \) into the function:

• \( f(0) = \log_{10}(3 - 0) = \log_{10}(3) \)

This calculation shows that the y-intercept of the graph is at the point \( (0, \log_{10}(3)) \). Here, \( \log_{10}(3) \) is a constant which gives the exact height at which the function crosses the y-axis.

Graphing techniques

When graphing the function \( f(x) = \log_{10}(3-x) \), we use several techniques to ensure accuracy and clarity. Having identified key features such as the domain, vertical asymptote, and y-intercept, these guide the shape of the graph.

Begin by plotting significant points and noting behaviors:

• Plot the y-intercept \( (0, \log_{10}(3)) \).
• Recognize the vertical asymptote at \( x = 3 \), indicating that as the graph approaches this line from the left, it drops sharply downward.
• Note that as \( x \) tends to negative infinity, the values of the function slightly increase because the function approaches \( \log_{10}(3) \).

Draw the curve:

• Start from the left of the graph, tracing the rise towards \( \log_{10}(3) \) but ensure it doesn't level, indicating a slight logarithmic curve.
• As you approach \( x = 3 \), show the curve dropping steeply toward minus infinity, reflecting the behavior near the asymptote.

Connect these dots smoothly to form the full graph. Graphing by considering the domain, asymptote, and intercept queues ensures your function accurately displays its behavior and relation to each axis.