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Chapter 4: Problem 48

Sketch the graph of \(f\). $$f(x)=\log _{2} \sqrt[3]{x}$$

Short Answer

Expert verified
The graph of \(f(x)=\log_{2} \sqrt[3]{x}\) passes through \((1,0)\), has a vertical asymptote at \(x=0\), and grows slowly to infinity as \(x\) increases.

Step by step solution

01

Understand the Function

The function given is \(f(x)=\log _{2} \sqrt[3]{x}\). This means we are dealing with a logarithmic function with base 2 of the cube root of \(x\). The cube root of \(x\) can be rewritten as \(x^{1/3}\). Therefore, the function can be expressed as \(f(x) = \frac{1}{3} \log_{2}(x)\).
02

Identify Key Features

For the function \(f(x) = \frac{1}{3} \log_{2}(x)\), it is only defined for \(x > 0\). There is no output for \(x \leq 0\). The logarithmic function always passes through the point where the argument of the log function is 1, that is, \(f(1) = \frac{1}{3} \log_{2}(1) = 0\). So, the graph will pass through the point \((1, 0)\).
03

Determine the Asymptote

As \(x\) approaches 0 from the positive side, \(\log_{2}(x)\) approaches \(-\infty\). Therefore, the graph has a vertical asymptote at \(x = 0\).
04

Analyze the Behavior at Infinity

As \(x\) approaches infinity, \(f(x)\) will also go to infinity, albeit slowly. This is because \(f(x) = \frac{1}{3} \log_{2}(x)\) grows logarithmically.
05

Sketch the Graph

Start the graphing at \((1, 0)\) since \(f(1) = 0\). Draw the curve to show the behavior moving towards \(-\infty\) as \(x\) approaches 0 from the positive side. Show that the curve grows slowly to \(\infty\) as \(x\) moves to \(\infty\). Ensure there is a vertical asymptote at \(x=0\).

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Key Concepts

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

Graphing Logarithmic Functions
Graphing a logarithmic function involves interpreting how the function behaves for different values of the variable. For the function \(f(x) = \frac{1}{3} \log_{2}(x)\), the key feature is that it only exists for \(x > 0\). You plot the logarithmic curve starting from the point where the argument of the logarithm equals 1, i.e., \((1, 0)\). This is because the logarithms of 1 is always zero.
  • Logarithmic graphs are defined on the positive x-axis.
  • The graph passes through the point where the logarithm's argument is 1.
  • The curve slowly rises in the positive x-axis direction.
The graph will look like a smooth curve starting from \((1, 0)\) and increasing towards infinity as \(x\) approaches infinity.
Asymptotes
An asymptote is a line that a graph approaches but never touches or crosses. In the case of the function \(f(x) = \frac{1}{3} \log_{2}(x)\), there is a vertical asymptote at \(x = 0\).
  • A vertical asymptote means that as \(x\) gets closer to 0, the value of \(f(x)\) decreases towards negative infinity.
  • Logarithmic functions commonly have vertical asymptotes where their argument is zero.
Understanding asymptotes helps ensure that you correctly sketch the behavior of the function near the edge of its domain. Always remember that the graph will keep close to but not actually reach the line \(x = 0\).
Cube Root
The cube root function \(\sqrt[3]{x}\) can be rewritten as \(x^{1/3}\). This transformation is helpful to better understand the behavior of the given logarithmic function. In this exercise, the expression \(\log_{2}(\sqrt[3]{x})\) simplifies to \(\frac{1}{3}\log_{2}(x)\).
  • The cube root is the same as raising \(x\) to the power of \(\frac{1}{3}\).
  • Knowing how to manipulate these expressions helps in simplifying logarithmic functions.
By reinterpreting cube roots as exponentials, you make it easier to handle complex expressions both analytically and when graphing.
Transformations of Functions
Transformations are modifications to the function's form, which shift, stretch, or compress the graph. For \(f(x) = \frac{1}{3} \log_{2}(x)\), this transformation is a vertical compression by a factor of \(\frac{1}{3}\).
  • Vertical transformations affect the graph's height.
  • Multiplying the logarithmic output by \(\frac{1}{3}\) flattens the growth rate.
Understanding transformations provides insights into how graphs change. For \(\log_{2}(x)\), a vertical compression makes the curve less steep. This helps in visualizing functions and predicting their behavior under different transformations, aiding in problem-solving and graph adjustment scenarios.

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Most popular questions from this chapter

Graph \(f\) on the given interval. (a) Estimate where \(f\) is increasing or is decreasing (b) Estimate the range of \(f\). $$f(x)=\frac{3.1^{-x}-4.1^{x}}{4.4^{-x}+5.3^{x}} ; \quad[-3,3]$$

Graph fin the given viewing rectangle. Use the graph of \(f\) to predict the shape of the graph of \(f^{-1}\). Verify your prediction by graphing \(f^{-1}\) and the line \(y=x\) in the same viewing rectangle. \(f(x)=\sqrt[3]{x-1} ; \quad[-12,12]\) by \([-8,8]\)

Graph \(f\) on the given interval. (a) Determine whether \(f\) is one-to-one. (b) Estimate the zeros of \(f\). $$f(x)=\pi^{0.6 x}-1.3^{\left(x^{1.8}\right)} ; \quad[-4,4]$$ (Hint: Change \(x^{1.8}\) to an equivalent form that is defined for \(x<0\).)

Exer. \(51-54\) : Chemists use a number denoted by \(p H\) to describe quantitatively the acidity or basicity of solutions. By definition, \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right],\) where \(\left[\mathbf{H}^{+}\right]\) is the hydrogen ion concentration in moles per liter. Many solutions have a pH between 1 and 14. Find the corresponding range of \(\left[\mathrm{H}^{+}\right]\)

Drug absorption If a 100 -milligram tablet of an asthma drug is taken orally and if none of the drug is present in the body when the tablet is first taken, the total amount \(A\) in the bloodstream after \(t\) minutes is predicted to be $$ A=100[1-(0.9)] \quad \text { for } \quad 0 \leq t \leq 10 $$ (a) Sketch the graph of the equation. (b) Determine the number of minutes needed for 50 milligrams of the drug to have entered the bloodstream.
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