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Chapter 3: Problem 63

Use a graphing utility to graph the function. $$f(x)=\log \sqrt[3]{x}$$

Short Answer

Expert verified
The graph of the function \(f(x) = \log \sqrt[3]{x}\) or equivalently \(f(x) = \frac{1}{3}\log x\) is the graph of a logarithmic function shifted such that it represents a logarithmic curve which starts near at (-鈭�, 0), passes through the point (1, 0), and then reaches upwards towards positive infinity as \(x\) increases. The function is only defined for positive values of \(x\).

Step by step solution

01

Understand the function

The function given by the exercise is \(f(x) = \log \sqrt[3]{x}\) This is a combination of the logarithmic function and the cube root function. As per the logarithmic property, this function can be reformulated as \(f(x) = \frac{1}{3}\log x\). The function is defined for all positive values of \(x\).
02

Calculate the function values

To plot this function accurately, it's recommended to calculate values of \(f(x)\) at various points. Here are some values:\n When \(x=0.1\), \(f(x) = -0.667\n When \(x=1\), \(f(x) = 0\n When \(x=10\), \(f(x) = 0.333\n When \(x=100\), \(f(x) = 0.667\
03

Graph the function

Use these values to accurately graph the function using a reliable graphing utility. The graph represents a logarithmic curve which starts near at (-鈭�, 0), passes through the point (1, 0), and then reaches upwards towards positive infinity as \(x\) increases.

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Key Concepts

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

Cube Root Function
Understanding cube root functions is crucial when working with transformations that involve roots. The cube root function, denoted as \( \sqrt[3]{x} \), provides numbers whose cube equals \(x\). Unlike square roots, cube roots can be calculated for negative numbers as well because the cube of a negative number is negative.
For example, \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \), and \( \sqrt[3]{-27} = -3 \) because \( (-3)^3 = -27 \).
The cube root transforms values across the full range of numbers (positive, zero, and negative), making it a versatile tool in mathematical transformations.
Logarithmic Properties
Logarithms convert multiplicative relationships into additive ones. In our problem, we deal with \( \log \sqrt[3]{x} \), and applying the property \( \log(a^b) = b\log a \), this can be rewritten as \( \frac{1}{3} \log x \).
  • Base: Typically the base 10 logarithm (common log) is used, unless specified as natural log (base \(e\)).
  • Domain: Log functions are only valid for positive \(x\) values, as logarithms of zero or negative numbers are undefined in the real number system.
  • Range: The range of \(\log x\) extends over all real numbers, turning multiplication into easier-to-handle addition or scaling.
This property of logarithms simplifies dealing with complex calculations and interpreting function transformations.
Graph Interpretation
Interpreting graphs involves understanding what changes occur in the function values over the \(x\)-axis. For our function \(f(x) = \frac{1}{3}\log x\), the graph:
  • Starts near negative infinity on the \(y\)-axis as \(x\) approaches 0 from the positive side.
  • Passes smoothly through the point (1, 0), where the log value is 0.
  • Rises steadily towards positive infinity as \(x\) grows larger.
This means, at small \(x\)-values, the function values are negative and increase gradually to zero at \(x = 1\), following an upward curve afterward.
Understanding the shape of a typical logarithmic curve helps in predictions and better analysis of changes alongside the \(x\)-axis.
x-values Calculation
Calculating \(x\)-values for a function helps outline key points in a graph. Here's how it applies to our logarithmic function:
First, choose a set of \(x\)-values over the desired range, such as \(0.1, 1, 10, \) and \(100\).
  • For \(x = 0.1\), plugging into the function gives \(f(x) = -0.667\).
  • At \(x = 1\), \(f(x) = 0\), which is a vital reference point as it represents zero crossing.
  • When \(x = 10\), \(f(x) = 0.333\), and for \(x = 100\), \(f(x) = 0.667\).
These calculated values are critical when graphing to depict the function's behavior accurately, highlighting the curve's trend and aiding in interpreting outcomes of the function over its domain.

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