Problem 267 - Calculus Volume 1

Question

Sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. \(f(x)=1-\ln x\)

Step-by-Step Solution

Answer

Domain: \((0, \infty)\), Range: \((-\infty, \infty)\), Vertical Asymptote: \(x = 0\).

Step 1: Identify the Function and Move Points

The function given is \( f(x) = 1 - \ln x \). This function is a transformation of the natural logarithmic function \( y = \ln x \). Here, every point on the graph of \( y = \ln x \) is reflected over the x-axis and shifted 1 unit up.

Step 2: Determine the Domain

The natural logarithm \( \ln x \) is only defined for \( x > 0 \). Therefore, the domain of \( f(x) = 1 - \ln x \) is the same as that of the original logarithmic function: \( x > 0 \). In interval notation, the domain is \((0, \infty)\).

Step 3: Determine the Range

The range of the function \( y = \ln x \) is \((-\infty, \infty)\). When reflecting it over the x-axis and moving it 1 unit up, the range remains \((-\infty, \infty)\), as \( f(x) = 1 - \ln x \) still reaches all real numbers through these transformations.

Step 4: Identify the Vertical Asymptote

Since the natural logarithmic function \( \ln x \) has a vertical asymptote at \( x = 0 \), the transformed function \( f(x) = 1 - \ln x \) also maintains a vertical asymptote at \( x = 0 \). This means the graph approaches this line but never touches or crosses it.

Step 5: Sketch the Graph

Create a set of key points from the parent function \( y = \ln x \), such as \((1, 0)\), \((e, 1)\), etc. Reflect these points over the x-axis and move them 1 unit up to \((1, 1)\), \((e, 0)\), etc. plot these points on a coordinate plane. Draw a smooth curve through these points recognizing the asymptotic behaviour at \( x = 0 \).

Key Concepts

Graph Sketching of Logarithmic Functions

When sketching the graph of a logarithmic function like \( f(x) = 1 - \ln x \), it is important to understand the parent function it originates from. The parent function in this case is \( y = \ln x \), which is a basic logarithmic curve that passes through the point \((1, 0)\) and continuously rises, albeit at a decreasing rate.

This particular function, \( f(x) = 1 - \ln x \), is a transformation of \( y = \ln x \). First, we reflect the parent function over the x-axis, turning the upward curve of \( y = \ln x \) downward. Then, we shift the resulting curve up by one unit. Key points to plot for sketching this would include moving the point \((1, 0)\) on \( y = \ln x \) to \((1, 1)\) on \( f(x) = 1 - \ln x \), and the point \((e, 1)\) to \((e, 0)\).

• Reflect over the x-axis
• Shift up by 1
• Re-plot points such as \((1, 1)\) and \((e, 0)\)
• Draw a smooth curve

This graph will appear to approach but never touch the y-axis, maintaining an asymptotic behavior.

Domain and Range of Logarithmic Functions

Understanding the domain and range of \( f(x) = 1 - \ln x \) is crucial for fully grasping the function's behavior.

Starting with the domain, logarithmic functions are only defined for positive real numbers. This is because you cannot take the logarithm of zero or a negative number. For \( y = \ln x \), the domain is \((0, \infty)\), and this remains true even after applying transformations to the function. Therefore, the domain of \( f(x) = 1 - \ln x \) is also \( x > 0 \) or, in interval notation, \((0, \infty)\).

• Logarithms require positive \( x \)
• Domain: \( x > 0 \)

As for the range, the original function \( y = \ln x \) covers all real numbers—everything from negative to positive infinity. When reflected over the x-axis and adjusted by a vertical shift of one unit, the range of the transformed function \( f(x) = 1 - \ln x \) remains the entire set of real numbers, or \((−\infty, \infty)\).

• Range remains \((−\infty, \infty)\)

Vertical Asymptotes of Logarithmic Functions

Vertical asymptotes are lines that the graph of a function approaches but never actually touches or crosses.

For the basic logarithmic function \( y = \ln x \), there is a vertical asymptote at \( x = 0 \). This is because as \( x \) gets closer to zero from the positive side, \( \ln x \) decreases without bound towards negative infinity. In the transformed function \( f(x) = 1 - \ln x \), this vertical asymptote persists. The transformation does not affect the position of the vertical asymptote, only the nature of the graph's behavior around it.

• Asymptote at \( x = 0 \)
• The graph never touches or crosses \( x = 0 \)
• Transformation does not move the asymptote

Recognizing the presence and nature of this asymptote is important when sketching or interpreting the graph, as it indicates the boundary of the function's domain.