/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 Sketch the graph of \(f\). $$f... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 45

Sketch the graph of \(f\). $$f(x)=\log _{2}\left(x^{3}\right)$$

Short Answer

Expert verified
The graph of \( f(x) = \log_{2}(x^3) \) is a logarithmic curve starting from \(-\infty\) near \( x=0 \), passing through (1,0), and increasing to \(\infty\).

Step by step solution

01

Identify the Domain

To find the domain of the function \( f(x) = \log_{2}(x^3) \), we need \( x^3 > 0 \), because the logarithm is only defined for positive numbers. This means \( x > 0 \). Thus, the domain of \( f(x) \) is \( x > 0 \).
02

Analyze Behavior at Critical Points

Consider special points to understand behavior. At \( x = 1 \), we have \( f(1) = \log_{2}(1^3) = \log_{2}(1) = 0 \). As \( x \to 0^+ \), \( x^3 \to 0^+ \), thus \( f(x) = \log_{2}(x^3) \to -\infty \).
03

Determine End Behavior

Evaluate the function as \( x \to \infty \). Since \( x^3 \to \infty \) as \( x \to \infty \), \( f(x) = \log_{2}(x^3) = 3 \cdot \log_{2}(x) \to \infty \). This indicates the graph of \( f(x) \) keeps increasing with no upper bound.
04

Sketch the Graph

Based on the analysis, sketch the graph: Start near the origin, approaching the vertical asymptote at \( x = 0 \), passing through the point (1,0), and continue to increase without bound as \( x \to \infty \). The graph is smooth and continuously increasing, reflecting the behavior of a logarithmic function.

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Key Concepts

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

Domain of a Function
In mathematics, the domain of a function refers to the set of all possible input values (usually represented by \(x\)) for which the function is defined. For the function \(f(x) = \log_{2}(x^3)\), understanding the domain is crucial because logarithmic functions are only defined for positive arguments. This means, for our specific function:
  • We need \(x^3 > 0\).
  • This simplifies to \(x > 0\) because any negative value of \(x\) cubed remains negative.
Hence, the domain of \(f(x)\) is all positive numbers: \(x > 0\). This restriction is essential to make sure our function is working with valid numbers that produce real results.
Asymptotic Behavior
The asymptotic behavior of a function describes how it behaves as it approaches certain limits or boundaries. For logarithmic functions like \(f(x) = \log_{2}(x^3)\), vertical asymptotes are common. Here:
  • When \(x\) approaches 0 from the positive side (denoted as \(x \to 0^+\)), \(x^3\) tends toward zero, meaning the input of the log approaches zero.
  • Since \(\log_{2}(x^3)\) tends to negative infinity as \(x^3\) approaches zero, there is a vertical asymptote at \(x = 0\).
This means on the graph, as \(x\) draws nearer to zero from the right, the value of \(f(x)\) drops sharply downward, representing this undefined yet approachable boundary.
Graphing Functions
When graphing a function, especially a logarithmic one like \(f(x) = \log_{2}(x^3)\), it involves plotting points that represent the function's values for various inputs of \(x\), then drawing a curve that smoothly connects these points.
  • Begin by noting the point it passes through known values: for example, \(f(1) = \log_{2}(1^3) = 0\).
  • From this point, we know the function starts at (1,0) on the graph.
  • The function increases continuously as \(x\) increases, with the curve rising to the right without bound.
The plotted graph provides a visual understanding of how \(f(x)\) behaves, showing clearly how it approaches the point (1,0) as a reference along its continuous ascent as \(x\) grows, reflecting the nature of logarithms.
End Behavior
End behavior in functions describes how a graph behaves as \(x\) goes towards infinity or negative infinity. For the function \(f(x) = \log_{2}(x^3)\):
  • As \(x \to \infty\), \(x^3\) also tends toward infinity.
  • Therefore, \(f(x) = \log_{2}(x^3)\) continues to rise without bound.
  • Explicitly, \(f(x) = 3 \log_{2}(x)\), indicating a continuously increasing curve.
This unbounded increase showcases that there is no horizontal asymptote 鈥� the function expands upwards forever, describing its asymptotic choice through infinity, a hallmark of logarithmic function expansion.

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Most popular questions from this chapter

Exer. \(43-46:\) Use natural logarithms to solve for \(x\) in terms of \(y\) $$y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$

Exer. \(47-48\) : Sketch the graph of \(f,\) and use the change of base formula to approximate the \(y\) -intercept. $$f(x)=\log _{3}(x+5)$$

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$M=\frac{L r k}{12(k-1)}$$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Business loan The owner of a small business decides to finance a new computer by borrowing 3000 dollars for 2 years at an interest rate of \(7.5 \%\) (a) Find the monthly payment. (b) Find the total interest paid on the loan.

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 \(\left[\mathrm{H}^{+}\right]\)

Graph \(f\) on the given interval. (a) Estimate the largest interval \([a, b]\) with \(a< 0< b\) on which \(f\) is one-to-one. (b) If \(g\) is the function with domain \([a, b]\) such that \(g(x)=f(x)\) for \(a \leq x \leq b,\) estimate the domain and range of \(g^{-1}.\) $$f(x)=2.1 x^{3}-2.98 x^{2}-2.11 x+3 ; \quad[-1,2]$$
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