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Chapter 1: Problem 264

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. $$ f(x)=3+\ln x $$

Short Answer

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

Step by step solution

01

Understanding the Function Form

The given function is \( f(x) = 3 + \ln x \). This is a transformation of the basic logarithmic function \( \ln x \). The function \( \ln x \) has a vertical asymptote at \( x = 0 \) and is only defined for \( x > 0 \). The transformation \( 3 + \ln x \) means the graph of \( \ln x \) is shifted up by 3 units.
02

Identifying the Domain

The domain of \( 3 + \ln x \) is the set of all \( x \) for which the function is defined. Since logarithmic functions are defined for positive \( x \) values, the domain of \( f(x) = 3 + \ln x \) is \( x > 0 \), or in interval notation, \((0, \infty)\).
03

Identifying the Range

Logarithmic functions typically have a range of all real numbers. The vertical shift does not affect this characteristic. Therefore, the range of \( 3 + \ln x \) remains as \( (-\infty, \infty) \).
04

Identifying the Vertical Asymptote

The vertical asymptote for a logarithmic function such as \( \ln x \) remains unchanged by vertical shifts. Hence, the vertical asymptote for \( 3 + \ln x \) is \( x = 0 \).
05

Sketching the Graph

To sketch the graph, start with the basic shape of \( \ln x \), which rises slowly and approaches the vertical asymptote at \( x = 0 \) from the right. Shift the entire graph upward by 3 units to account for the transformation \( 3 + \ln x \). Ensure to clearly mark the asymptote at \( x = 0 \) and label the axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Domain of a Logarithmic Function
The domain of a function describes all the possible input values (or x-values) for which the function is defined. For logarithmic functions like \( f(x) = \ln{x} \), they are only defined for positive values of \( x \). This is because you cannot take the logarithm of a negative number or zero in the real number system.
The given function \( f(x) = 3 + \ln{x} \) maintains this fundamental property of the logarithm. Therefore, the domain remains \( x > 0 \).
  • In interval notation, this domain is represented as \((0, \infty)\).
  • This means that any positive value can be plugged into the function, and it will yield a result.
When interpreting or sketching the graph, remember that it will not touch or cross the \( y \)-axis, as the function is undefined at \( x = 0 \) and any negative \( x \) values.
Exploring the Range of a Logarithmic Function
The range of a function is all the possible output values or y-values that come out of the function. Logarithmic functions inherently have a wide range, stretching from \(-\infty\) to \(\infty\).
Even though our function is \( f(x) = 3 + \ln{x} \), where the graph is vertically shifted by 3 units, this does not alter the nature of its range. The vertical shift only affects the graph's position along the y-axis, not the set of outputs it can generate.
  • This means the range of \( f(x) = 3 + \ln{x} \) remains as \((-\infty, \infty)\).
  • The function can output practically any real number depending on the input value of \( x \).
When sketching the graph, you will see that it continues upwards and downwards indefinitely, without any horizontal asymptotes limiting the range.
Identifying Vertical Asymptotes in Logarithmic Functions
Vertical asymptotes in functions are vertical lines that the graph approaches but never actually touches or crosses. In logarithmic functions like \( \ln{x} \), a vertical asymptote occurs because the function is undefined at certain points.
For \( f(x) = 3 + \ln{x} \), the vertical asymptote is at \( x = 0 \). This is because logarithmic functions are not defined for zero or negative values.
  • The line \( x = 0 \) is a visual boundary that the graph of the function will edge closer to but never meet or cross.
  • This remains unchanged even when you apply vertical transformations, such as the addition of 3 in this case.
When you plot or view the graph, keep in mind that the curve will seem to get infinitely close to this asymptote from the right side, which visually enforces the idea of the function being undefined at \( x = 0 \).

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Most popular questions from this chapter

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