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Chapter 7: Problem 11

Graph the functions and identify their domains. $$f(x)=\ln 3 x$$

Short Answer

Expert verified
The domain of \(f(x) = \ln(3x)\) is \(x > 0\). The graph is a curve that starts from near negative infinity when \(x\) is just greater than 0 and increases as \(x\) increases. A vertical asymptote exists at \(x = 0\).

Step by step solution

01

Identify the Domain of the Function

The domain of the function \(f(x) = \ln(3x)\) is the set of all possible input values for \(x\) that result in valid outputs. The natural logarithm function, \(\ln(x)\), is defined only for positive real numbers. Therefore, for \(3x\) to be positive, \(x\) must be greater than 0. Hence, the domain of \(f(x)\) is \(x > 0\).
02

Plot the Graph

To graph the function, let's plot a few key points. For this, take values of \(x\) that are easy to compute and that fall within the domain of the function. For example: \(x_1 = 1/3\) yields \(f(x_1) = \ln(1) = 0\), \(x_2 = 1\) results in \(f(x_2) = \ln(3)\), and \(x_3 = 10/3\) gives \(f(x_3) = \ln(10)\). Then, draw a curve passing through these points, which increases as \(x\) increases, showing the natural logarithmic growth. The curve should approach the y-axis but never touch it, as \(x\) approaches 0.
03

Draw Asymptote

The function has a vertical asymptote at \(x = 0\) because as \(x\) approaches 0 from the right, the value of the natural logarithm function \(f(x)\) decreases without bound, approaching negative infinity. Thus, the graph will get infinitely close to the y-axis, but will not cross it or touch it.

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Key Concepts

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

Domain of a Function
Understanding the domain of a function is crucial for graphing and analyzing the function's behavior. The domain comprises all the possible values of the independent variable, usually denoted as 'x', for which the function is defined and produces real values.

For logarithmic functions such as the natural logarithm, their domain is restricted to positive real numbers because you cannot take the logarithm of a non-positive number. In the case of the function \(f(x) = (3x)\), the domain includes any value of 'x' where \(3x > 0\), because that ensures the argument of the natural logarithm is positive. Therefore, the domain is \(x > 0\), because any value of 'x' greater than zero will make '3x' a positive number, hence providing a real output.

Remember, identifying the domain is a fundamental step before graphing because it tells us the interval on the x-axis over which we will draw the function's graph.
Natural Logarithm
The natural logarithm is a special logarithmic function that鈥檚 denoted as \((x)\), where 'e' is the base of the natural log, roughly equal to 2.71828. It is a unique and extensively used function in mathematics, especially in calculus, and is the inverse function of exponential growth with base 'e'.

When a function is expressed as \(f(x) = (3x)\), it means that for every 'x' in its domain, you multiply it by 3 and then find the natural log of the result.

Properties of the Natural Logarithm

  • \((1) = 0\) because any number raised to the power of 0 is 1.
  • \((e) = 1\) because 'e' is the base of the natural logarithm.
  • The function is continuous and increases slowly as 'x' gets larger.
  • The graph of the natural logarithm never touches the negative x-axis, yet continues indefinitely to the right.
These properties are reflected in the graph of any function involving the natural logarithm, such as the one given in the exercise.
Vertical Asymptote
A vertical asymptote on a graph is a vertical line that the function approaches but never actually reaches or crosses. It signifies that the value of the function grows without bound in either the positive or negative direction as it gets closer to the line.

In the context of the natural logarithm, a vertical asymptote occurs at \(x = 0\) because as the input values get infinitesimally close to zero from the right, the natural logarithm of those values decreases without bound鈥攈ead towards negative infinity.

Significance of Vertical Asymptote

  • It delineates the boundary beyond which the function is not defined鈥攊n this case, to the left of the y-axis.
  • During graphing, it serves as a guide for sketching the curve, as the graph will approach this line but not cross it.
The vertical asymptote is a powerful concept that helps us understand the limits and behavior of functions near points where they are not defined.

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Most popular questions from this chapter

Demonstrate numerically the propertics of logarithms. \(\log 2^{5}=5 \log 2\)

Determine whether the data has the add-add, add-multiply, multiply-multiply, or constant-second-differences pattern. Identify the type of function that has the pattern. $$\begin{array}{rr} x & f(x) \\ \hline 2 & 4.6 \\ 4 & 6.0 \\ 6 & 7.4 \\ 8 & 8.8 \\ 10 & 10.2 \end{array}$$

Shows the cubic function $$f(x)=x^{3}-6 x^{2}+5 x+20$$ a. Make a table of values of \(f(x)\) for each integer value of \(x\) from 1 to 6 b. Show that the third differences between the values of \(f(x)\) are constant. You can calculate the third differences in a time-efficient way using the list and delta list features of your grapher. If you do it by pencil and paper, be sure to subtract from a function value, the previous value in each case. c. Make a conjecture about how you could determine whether a quartic function (fourth degree) fits a set of points.

Determine whether the data has the add-add, add-multiply, multiply-multiply, or constant-second-differences pattern. Identify the type of function that has the pattern. $$\begin{array}{rr} x & f(x) \\ \hline 1 & 25 \\ 5 & 85 \\ 9 & 113 \\ 13 & 109 \\ 17 & 73 \end{array}$$

Graph the functions and identify their domains. $$f(x)=\log _{3} x^{2}$$
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