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Chapter 5: Problem 25

Graph each function and specify the domain, range, intercept(s), and asymptote. (a) \(y=\log _{2} x\) (b) \(y=-\log _{2} x\) (c) \(y=\log _{2}(-x)\) (d) \(y=-\log _{2}(-x)\)

Short Answer

Expert verified
Functions have domains as follows: (a) \( x > 0 \), (b) \( x > 0 \), (c) \( x < 0 \), (d) \( x < 0 \); all have a range of all real numbers.

Step by step solution

01

Identify the basic properties of the logarithmic function

For each logarithmic function, the basic form is given by \( y = \log_b(x) \). The domain is \( x > 0 \) because logarithms are only defined for positive values of \( x \). The range is all real numbers, \( \mathbb{R} \). The vertical asymptote is at \( x = 0 \). There is no horizontal asymptote, and the y-intercept occurs at \( (1, 0) \) if the base \( b \) is greater than 1.
02

Analyze the function \( y = \log_{2} x \)

**Domain**: \( x > 0 \), since the logarithm is undefined for non-positive \( x \). **Range**: All real numbers, \( \mathbb{R} \). **Vertical Asymptote**: \( x = 0 \). **Intercept**: Intersection with the y-axis at \( (1, 0) \). Graph: An increasing curve going from the third quadrant to the first, passing through \( (1,0) \). There is no horizontal intercept.
03

Analyze the function \( y = -\log_{2} x \)

**Domain**: \( x > 0 \). **Range**: All real numbers, \( \mathbb{R} \). **Vertical Asymptote**: \( x = 0 \). **Intercept**: Intersection with the y-axis at \( (1, 0) \). Graph: A decreasing curve, mirrored downward relative to \( y = \log_{2} x \), crossing \( (1, 0) \) and extending toward negative infinity as \( x \to \infty \).
04

Analyze the function \( y = \log_{2}(-x) \)

**Domain**: \( x < 0 \), as we must have \( -x > 0 \). **Range**: All real numbers, \( \mathbb{R} \). **Vertical Asymptote**: \( x = 0 \). **Intercept**: There is no intercept with the y-axis since no positive x satisfies the function. Graph: A reflection of \( y = \log_{2} x \) over the y-axis, decreasing from the second to the third quadrant.
05

Analyze the function \( y = -\log_{2}(-x) \)

**Domain**: \( x < 0 \). **Range**: All real numbers, \( \mathbb{R} \). **Vertical Asymptote**: \( x = 0 \). **Intercept**: No intercept with the y-axis, similar to \( y = \log_{2}(-x) \). Graph: A reflection of \( y = -\log_{2} x \) over the y-axis, increasing from the second to the fourth quadrant.

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Key Concepts

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

Function Domain
The domain of a function is a fundamental concept that defines all the possible input values, or x-values, for which the function is defined. For logarithmic functions, this domain is particularly crucial because logarithms are only defined for positive inputs when considering their basic form.
  • In the function \( y = \log_{2} x \), the domain is all positive real numbers: \( x > 0 \). This is because the logarithmic function becomes undefined at and below zero.
  • For \( y = \log_{2}(-x) \), the domain shifts to negative real numbers: \( x < 0 \). Here, \(-x\) needs to be positive for the logarithm to work, thus reversing the sign of the input condition.
For other transformations of the logarithmic function, such as negations or reflections, the domain remains similarly defined by the condition \( x > 0 \) or \( x < 0 \), dependent on whether the input is negated. Understanding the domain ensures you only choose x-values that will produce meaningful results in a logarithmic equation.
Function Range
The range of a function includes all the possible output values that a function can produce. In the case of logarithmic functions, despite restrictions on the input domain, the range remains unrestricted.
  • For each function analyzed, \( y = \log_{2} x \), \( y = -\log_{2} x \), \( y = \log_{2}(-x) \), and \( y = -\log_{2}(-x) \), the range is all real numbers: \( \mathbb{R} \).
The reason the range is \( \mathbb{R} \) is due to the nature of a logarithm's curve, which gradually extends upwards or downwards indefinitely, depending on the transformation applied. This means that as the x-values get larger or smaller, the y-values will stretch infinitely, covering all possible real numbers.
Graphing Functions
Graphing logarithmic functions offers a visual representation of the relationship between x and y, helping to comprehend the function鈥檚 behavior clearly. For the basic \( y = \log_{2} x \), the graph is a smooth curve that increases steadily, starting from negative infinity to positive infinity.Within the graph:
  • \( y = \log_{2} x \) starts from negative infinity and gradually increases through the point \((1, 0)\) since \( \log_{2} 1 = 0 \), and rises continually as \( x \) increases.
  • \( y = -\log_{2} x \) is a reflection over the x-axis, moving downward yet passing through \((1, 0)\) as well.
  • For graphs like \( y = \log_{2}(-x) \) and \( y = -\log_{2}(-x) \), the curves assume their mirror image, displaying reflections over the y-axis, indicating that they have negative x-values in their domain.
Through plotting these functions, their characteristic vertical asymptote at \( x = 0 \) and the absence of any horizontal intercepts become clearly visible.
Asymptotes
Asymptotes play a vital role in understanding the behavior of logarithmic functions at boundaries. They indicate the line that the graph approaches but typically never touches. For logarithmic functions:
  • An important vertical asymptote occurs at \( x = 0 \). This is because, as x approaches zero from the right for functions like \( y = \log_{2} x \), the value of y heads toward negative infinity, creating a boundary it approaches but cannot cross.
  • Similarly, functions with a negative inside, such as \( y = \log_{2}(-x) \), maintain their vertical asymptote at \( x = 0 \), because they graph on the negative x-axis, advancing toward the asymptotes in a mirrored manner compared to their positive x counterparts.
Understanding asymptotes aids in sketching graphs since they provide critical insight into the function's limits and boundaries, essential for fully capturing the nature of a logarithmic function's graph.

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Most popular questions from this chapter

Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximation. $$\log _{2} \frac{2 x-1}{x-2}<0$$

Decide which of the following properties apply to each function. (More than one property may apply to a function.)A. The function is increasing for \(-\infty

Let \(\mathcal{N}=\mathcal{N}_{0 e^{k t}} .\) In this exercise we show that if \(\Delta t\) is very small, then \(\Delta \mathcal{N} / \Delta t \approx k \mathcal{N} .\) In other words, over very small intervals of time, the average rate of change of \(\mathcal{N}\) is proportional to \(\mathcal{N}\) itself. (a) Show that the average rate of change of the function \(\mathcal{N}=\mathcal{N}_{0} e^{t t}\) on the interval \([t, t+\Delta t]\) is given by $$\frac{\Delta \mathcal{N}}{\Delta t}=\frac{\mathcal{N}_{0} e^{k t}\left(e^{k \Delta t}-1\right)}{\Delta t}=\frac{\mathcal{N}\left(e^{k \Delta t}-1\right)}{\Delta t}$$ (b) In Exercise 26 of Section 5.2 we saw that \(e^{x} \approx x+1\) when \(x\) is close to zero. Thus, if \(\Delta t\) is sufficiently small, we have \(e^{k \Delta t} \approx k \Delta t+1 .\) Use this approximation and the result in part (a) to show that \(\Delta \mathcal{N} / \Delta t \approx k N\) when \(\Delta t\) is sufficiently close to zero.

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