Problem 28 - Calculus with Concepts in Calculus

Question

Sketch the graph of \(f\). $$ f(x)=\ln (e x) $$

Step-by-Step Solution

Answer

The graph of \( f(x) = 1 + \ln x \) shifts \( \ln x \) up by 1 and is defined for \( x > 0 \). It passes through \( (1, 1) \) with a vertical asymptote at \( x = 0 \).

Step 1: Simplify the Function

Start by simplifying the function. We have \[ f(x) = \ln (e x) \]Using the property of logarithms, \(\ln (ab) = \ln a + \ln b\), we can simplify this as:\[ f(x) = \ln e + \ln x = 1 + \ln x \]Now our function to sketch is \( f(x) = 1 + \ln x \).

Step 2: Determine Domain and Range

The function \( \ln x \) is only defined for \( x > 0 \). Since \( \ln x \) is part of our function, the domain of \( f(x) = 1 + \ln x \) is also \( x > 0 \).The range of \( \ln x \), which extends from \(-\infty\) to \(\infty\), is the same for \( f(x) = 1 + \ln x \) since adding 1 only shifts the graph vertically.

Step 3: Identify Key Features

Identify key features of the logarithmic function. The natural logarithm \( \ln x \) crosses the x-axis at \( x=1 \) so our function \( f(x) = 1 + \ln x \) will cross the x-axis higher by 1 unit because of the \(+1\) shift. Therefore, the point is \( (1, 1) \).Also, as \( x \to 0^+ \), \( \ln x \to -\infty \), and as \( x \to \infty \), \( \ln x \to \infty \), hence \( f(x) \to \infty \) as \( x \to \infty \) and \( f(x) \to -\infty \) as \( x \to 0^+ \).

Step 4: Plot Points and Sketch the Graph

Plot several points to understand the graph shape. For example, calculate \( f(x) \) for a few values:- \( f(0.5) = 1 + \ln(0.5) \approx 0.307 \)- \( f(1) = 1 + \ln(1) = 1 \)- \( f(e) = 1 + \ln(e) = 2 \)Using these points, sketch the graph. The graph should have a vertical asymptote at \( x = 0 \) and increase from left to right.

Step 5: Label the Graph

Ensure the graph is well-labeled. Mark the point \((1, 1)\) as a reference point, as well as any other calculated points. Indicate the asymptote at \( x = 0 \). Remember, the graph slowly approaches the asymptote but never crosses it, reflecting the domain \( x > 0 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and have the form \( f(x) = \ln(x) \), where \( \ln \) denotes the natural logarithm. The base of the natural logarithm is \( e \), approximately equal to 2.718. A key property of logarithmic functions is that they are only defined for positive values of \( x \). This means you can only find the logarithm of a positive number. Logarithmic functions are widely used in various fields such as science, finance, and computer science due to their unique properties. They grow slowly compared to polynomial or exponential functions, making them useful in modeling situations where growth diminishes over time.Understanding the properties of the logarithmic function is essential when dealing with calculations and transformations. For instance, knowing that \( \ln(e) = 1 \) helps simplify functions like \( f(x) = \ln(ex) = 1 + \ln(x) \). This simplification stems from the identity \( \ln(a \cdot b) = \ln(a) + \ln(b) \), allowing us to break down complex logarithmic expressions into simpler components.

Function Transformation

Transformations of functions describe how various manipulations, such as translations, reflections, or stretches, affect a function's graph. With logarithmic functions, a transformation often involves shifting the graph up, down, left, or right.The function \( f(x) = 1 + \ln(x) \) is a transformed version of \( \ln(x) \). The addition of 1 represents a vertical shift upwards by 1 unit. In function transformations:

• Vertical shifts occur from adding/subtracting constants to the function \((y = f(x) + c)\).
• Horizontal shifts come from adding/subtracting constants inside the argument \((y = f(x + c))\).
• Reflections occur through multiplying by negative factors \((y = -f(x))\).
• Stretches/compressions happen by multiplying the variable or function by constant factors.

Understanding these transformations allows us to predict how the changes will manifest graphically and dynamically see the effects of simple adjustments to the functions.

Domain and Range

The domain and range form the foundation of understanding functions. They define what input \((x)\) and output \((y)\) values a function can handle or produce. For logarithmic functions like \( f(x) = \ln(x) \), the domain is limited to positive numbers \((x > 0)\) because \( \ln(x) \) isn't defined for zero or negative values. This constraint is carried over to any transformed function, such as \( f(x) = 1 + \ln(x) \), resulting in the domain also being \( x > 0 \).The range of a logarithmic function is all real numbers \((-\infty, \infty)\). This is because, as \( x \) increases, \( \ln(x) \) can reach any negative or positive number. Adding a constant, like in \( 1 + \ln(x) \), does not alter this infinite range but rather shifts it vertically. Understanding these constraints helps in sketching accurate graphs and predicting the behavior of functions at extreme values.

Vertical Asymptote

A vertical asymptote is a line \( x = a \) where a function's graph approaches but never quite touches or crosses. In logarithmic functions, vertical asymptotes often occur due to restrictions in the domain. For \( f(x) = 1 + \ln(x) \), the vertical asymptote is at \( x = 0 \).As \( x \) approaches zero from the positive side, \( \ln(x) \) tends towards negative infinity \((-\infty)\). This means the graph of \( 1 + \ln(x) \) also approaches \(-\infty\), reinforcing that \( x = 0 \) is an asymptote. Key characteristics of vertical asymptotes:

• Functions increase or decrease without bound as they near the asymptote.
• Vertical asymptotes highlight boundaries in a function's domain.

Recognizing vertical asymptotes allows us to accurately describe a function's behavior around crucial points where dramatic changes occur.