Problem 264 - Calculus Volume 1

Question

Sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. \(f(x)=3+\ln x\)

Step-by-Step Solution

Answer

Domain: \(x > 0\); Range: \((-\infty, \infty)\); Vertical asymptote: \(x = 0\).

Step 1: Understand the Function

The function given is \( f(x) = 3 + \ln x \). This is a vertical transformation of the natural logarithm function \( \ln x \). The \( \ln x \) graph is shifted up by 3 units.

Step 2: Determine the Domain

The domain of the natural logarithm function \( \ln x \) is \( x > 0 \). Therefore, the domain of \( f(x) = 3 + \ln x \) is also \( x > 0 \).

Step 3: Identify the Range

The range of \( \ln x \) is all real numbers, \( (-\infty, \infty) \). Since we are just shifting the graph up, the range of \( f(x) = 3 + \ln x \) remains the same, which is also \( (-\infty, \infty) \).

Step 4: Locate the Vertical Asymptote

The natural logarithm function \( \ln x \) has a vertical asymptote at \( x = 0 \). This asymptote remains unchanged even after shifting the function vertically by adding 3. So, the vertical asymptote for \( f(x) \) is \( x = 0 \).

Step 5: Sketch the Graph

Begin by sketching the basic graph of \( \ln x \), which passes through \( (1,0) \). Shift the entire graph up 3 units so it passes through \( (1,3) \). The general shape, with a slow rise to the right and a steep drop off approaching \( x = 0 \), remains. Remember the vertical asymptote at \( x = 0 \).

Key Concepts

Graph Transformations

In mathematics, graph transformations involve moving or changing the shape and position of a graph on a coordinate plane. When we consider the function \( f(x) = 3 + \ln x \), we perform a vertical transformation on the graph of the natural logarithm function \( \ln x \). This specific transformation involves shifting the graph upwards by 3 units.

Such vertical transformations can be understood as altering the y-values of the graph without impacting its overall shape.

• If a constant is added to the function (as seen here with 3), the entire graph moves up.
• If a constant were subtracted, the graph would move downwards instead.

This transformation does not affect the x-values at all, which means it doesn't change the domain of the function.

Vertical Asymptote

A vertical asymptote is a vertical line that a graph approaches but never really touches. For the logarithmic function, \( \ln x \), the vertical asymptote is located at \( x = 0 \). This means that as \( x \) gets closer to 0 from the right side, the value of \( \ln x \) heads towards negative infinity.

In our function \( f(x) = 3 + \ln x \), even with the vertical transformation of moving the graph up by 3 units, the location of the vertical asymptote does not change—it remains at \( x = 0 \). This is because the transformation only alters the graph vertically, not horizontally.

• The asymptote acts as a boundary without limit.
• The graph never crosses this boundary but simply gets very close to it.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for that function. For the natural logarithm function \( \ln x \), the domain is \( x > 0 \) because logarithms are defined only for positive numbers.

Thus, for \( f(x) = 3 + \ln x \), the domain remains unchanged: it is still \( x > 0 \). In contrast, the range of a function is all the possible output values (y-values). For \( \ln x \), the range is all real numbers \((-\infty, \infty)\). Even after shifting the graph upwards by 3 units, the range does not change—it remains the entire set of real numbers.

• Domain: \( x > 0 \)
• Range: \((-\infty, \infty)\)
• These characteristics make logarithmic functions unique in terms of their unbounded output.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a logarithm with a base of \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It is a vital function in mathematics due to its applications in calculus, science, and engineering.

The function \( \ln x \) grows very slowly. As \( x \) increases, \( \ln x \) also increases but at a decreasing rate. It illustrates exponential growth but in inverse form.

• \( \ln 1 = 0 \) (since \( e^0 = 1 \))
• \( \ln x \) turns negative for \( 0 < x < 1 \)

Understanding how the natural logarithm works helps in comprehending the arithmetic of growth processes, like population dynamics, radioactivity, and compound interest.