Problem 48 - Calculus: Early Transcendentals

Question

(a) Graph the function \( f(x) = e^x + \ln | x - 4 | \) for \( 0 \le x \le 5 \). Do you think the graph is an accurate representation of \( f \)? (b) How would you get a graph that represents \( f \) better?

Step-by-Step Solution

Answer

The graph needs adjustments near x = 4 and zooming techniques to represent the function accurately.

Step 1: Understand the Function Components

The function consists of two parts: the exponential function \( e^x \) and the natural logarithm \( \ln |x - 4| \). Both of these functions have specific behaviors we need to consider. \( e^x \) increases rapidly for positive values of \( x \), while \( \ln |x - 4| \) has a vertical asymptote at \( x = 4 \). Our task is to evaluate this function within the interval \( 0 \le x \le 5 \).

Step 2: Analyze Domain Restrictions

For the function \( f(x) = e^x + \ln | x - 4 | \), both \( e^x \) and \( \ln | x - 4 | \) must be defined. Specifically, \( |x - 4| \) cannot be zero, which implies \( x eq 4 \). Therefore, the function is undefined at \( x = 4 \) and has a discontinuity there.

Step 3: Graph Function Components

Graph \( e^x \) independently to visualize its exponential growth. For \( \ln |x - 4| \), note the asymptote at \( x = 4 \), where the function approaches infinity as \( x \) approaches 4 from the right and negative infinity as \( x \) approaches 4 from the left. Plot these components on the same graph from \( x = 0 \) to \( x = 5 \).

Step 4: Evaluate Combined Graph

Combine the graphs of \( e^x \) and \( \ln |x - 4| \). Notice the effect that the ln term has on the graph: there's a notable spike at \( x = 4 \). Evaluate how this impacts the overall graph shape and note inaccuracies around \( x = 4 \) due to the discontinuity.

Step 5: Improve Graph Representation

To better represent \( f \), consider narrowing the range around the discontinuity at \( x = 4 \) to better visualize behavior on either side of the asymptote. Using a smaller range or zooming in on the region near the asymptote can improve clarity. Additionally, technology like graphing software can refine plots for better detail.

Step 6: Conclusion

By better handling the range around the asymptote, using graphing tools, and understanding the discontinuity, the graph represents \( f(x) = e^x + \ln | x - 4 | \) more accurately. Recognize this adjustment enhances interpretation of behavior around \( x = 4 \).

Key Concepts

Exponential Function

The exponential function is a fundamental mathematical function, denoted as \( e^x \). It is defined as the function where the constant \( e \) (approximately 2.718) is raised to the power of \( x \). This function has a few important characteristics:

• Rapid Growth: \( e^x \) increases quickly as \( x \) becomes larger. For positive values of \( x \), the exponential function grows exponentially. This means it gets bigger very fast as you move along the x-axis to the right.
• Positive Output: For all real numbers \( x \), the output of \( e^x \) is always positive.
• Non-Zero: Since \( e^x \) is positive, it never equals zero, which is vital when graphing or solving equations.

Understanding how \( e^x \) behaves is crucial when analyzing complex functions involving exponentials, as is the case with our given function \( f(x) = e^x + \ln |x - 4| \). The exponential segment heavily influences the shape of the graph for values of \( x \) far from any discontinuities.

Natural Logarithm

The natural logarithm, represented as \( \ln(x) \), is the inverse operation of the exponential function \( e^x \). It is particularly interesting when applied to expressions like \( \ln |x - 4| \). Here are its key features:

• Inverse Relationship: The natural logarithm \( \ln(x) \) undoes the exponential growth of \( e^x \). It determines how many times you'd multiply \( e \) by itself to achieve \( x \).
• Undefined for Non-Positive Values: \( \ln(x) \) is not defined for \( x \leq 0 \). For our function, this means \( \ln |x - 4| \) creates a constraint where the expression inside the logarithm must be positive.
• Outputs Negative Values: For \( 0 < x < 1 \), logarithmic outputs are negative. This affects the graph significantly near the point of discontinuity.

When exploring graphs and solving equations, particularly in calculus graphing, understanding how to manipulate and interpret \( \ln |x - 4| \) is essential, as it offers insight into the behavior of values of \( x \) near 4.

Discontinuity

A discontinuity is a point on a graph where the function is not defined, leading to a "break" or "gap" in the plotting of the function. In the context of \( f(x) = e^x + \ln | x - 4 | \):

• Point of Discontinuity: The function is undefined at \( x = 4 \). At this point, the natural logarithm's requirement that \( x - 4 eq 0 \) directly causes the discontinuity.
• Impact on Graph: As \( x \) approaches 4, from either direction, the value of \( \ln |x - 4| \) escalates towards infinity or negative infinity, causing the graph to make sharp turns.
• Visualizing: To accurately visualize functions with discontinuities, focus on graphing smoothly on either side of these points. Utilize graphing software to better handle these abrupt changes in behavior.

By understanding discontinuities, students can better interpret complex function behavior near these critical points.

Asymptote

An asymptote is a line that a graph approaches but never quite reaches. For the function \( f(x) = e^x + \ln |x - 4| \), there is a particularly poignant asymptote you must consider:

• Vertical Asymptote: This occurs at \( x = 4 \). The function \( \ln |x - 4| \) approaches infinity as \( x \) nears 4 from the right, and negative infinity from the left, implying a vertical asymptote.
• Graph Behavior: As \( x \) approaches the asymptote, the function’s value increases or decreases dramatically without actually reaching a finite value.
• Importance in Graphing: Recognizing where asymptotes lie helps accurately interpret and sketch the shape of a graph. Asymptotic behavior implies potential behavior not immediately obvious from equations.

With asymptotes, anticipate the direction and behavior of the graph around critical values, allowing for a precise depiction of the mathematical relationship.