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Magazine Chapter 5: Problem 56
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
Short Answer
Expert verified
F, G, and D do not apply; H and E apply. A applies as function increases in each domain part.
Step by step solution
01
Analyze the Function Type
The given function is \( y = \ln |x| \). This is a logarithmic function, and particularly, it involves the natural logarithm of the absolute value of \( x \). It is not a polynomial function as polynomials have the form \( a_nx^n + a_{n-1}x^{n-1} + \, ... + \, a_0 \) with powers of \( x \), not logarithms.
02
Determine the Domain
The domain of \( y = \ln |x| \) includes only the x-values for which the expression inside the logarithm is positive. Since it is the logarithm of an absolute value, \( x \) must not be zero, leading to a domain of \( (-\infty, 0) \cup (0, \infty) \).
03
Determine the Range
The range of any logarithmic function is \((-\infty, \infty)\) since a logarithm can produce any real number given suitable inputs. Thus, \( y = \ln |x| \) also has a range of \((-\infty, \infty)\).
04
Evaluate Continuity and Asymptotes
As \( x \to 0^+ \) or \( x \to 0^- \), \( \ln |x| \) tends to \(-\infty \), resulting in a vertical asymptote at \( x = 0 \). This function thus has an asymptote.
05
Determine Monotonic Behavior
For \( x > 0 \), \( \ln(x) \) is an increasing function, making \( \ln(|x|)\) increasing for \( x > 0 \). For \( x < 0 \), consider \( \ln(-x) \) which is decreasing as described by the transformation \( x \to -x \), thus \( \ln(|x|)\) is increasing for \( x < 0 \). The function is therefore continuously increasing in its entire domain, \((0, \infty)\) and \((-\infty, 0)\).
06
Evaluate Turning Points
A function has a turning point where the derivative changes sign. Since \( y = \ln |x| \) is increasing on both \(-\infty, 0\) and \(0, \infty\), there is no change in the sign of the derivative, and thus there are no turning points.
07
Evaluate One-to-One Property
A one-to-one function is a function where each \( y \) value has exactly one \( x \) value. Since each section of \( y = \ln |x| \), both for \( x > 0 \) and \( x < 0 \), is strictly increasing, the function is not one-to-one considering the entire real domain excluding 0.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain and Range
Understanding the domain and range of a function helps define where it operates and the results it can produce. For the function \( y = \ln |x| \), its domain is determined by the values of \( x \) that keep the expression inside the logarithm valid. Since \( \ln |x| \) requires \( x eq 0 \), the domain becomes \((-\infty, 0) \cup (0, \infty)\). This allows all positive and negative \( x \) except zero.
The range of \( y = \ln |x| \) spans all real numbers, or \((-\infty, \infty)\), because a logarithm can output any real value depending on the input. Essentially, the function can grow without bounds both positively and negatively, creating an unrestricted set of outputs.
The range of \( y = \ln |x| \) spans all real numbers, or \((-\infty, \infty)\), because a logarithm can output any real value depending on the input. Essentially, the function can grow without bounds both positively and negatively, creating an unrestricted set of outputs.
Asymptotes
Asymptotes are lines that a function's graph approaches but never reaches. They provide insight into the behavior and limits of a function. For \( y = \ln |x| \), there's a vertical asymptote at \( x = 0 \). This is because as \( x \) approaches zero from either the positive or negative side, \( \ln |x| \) trends towards \(-\infty\).
Vertical asymptotes indicate a point on the curve where the function shoots up to infinity or dives to negative infinity. For logarithmic functions like \( y = \ln |x| \), these points are significant because they highlight the boundary beyond which the function cannot extend.
Vertical asymptotes indicate a point on the curve where the function shoots up to infinity or dives to negative infinity. For logarithmic functions like \( y = \ln |x| \), these points are significant because they highlight the boundary beyond which the function cannot extend.
One-to-One Functions
A one-to-one function is where each output corresponds to exactly one input. This property is crucial for functions that need to have unique inverse functions.
For \( y = \ln |x| \), the function isn't one-to-one over its entire domain. While each segment \((x > 0)\) or \((x < 0)\) is strictly increasing, combining both segments over the whole domain breaks the one-to-one correspondence since you can't map every \( y \) back to a single \( x \).
For \( y = \ln |x| \), the function isn't one-to-one over its entire domain. While each segment \((x > 0)\) or \((x < 0)\) is strictly increasing, combining both segments over the whole domain breaks the one-to-one correspondence since you can't map every \( y \) back to a single \( x \).
- One-to-one functions pass the horizontal line test: no horizontal line intersects the function more than once.
- Since \( y = \ln |x| \) fails this test over both segments viewed together, it's not classically one-to-one.
Polynomial Functions
Polynomial functions are composed of variables raised to whole number powers combined with constant coefficients. Their structure is simple, following the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \). This allows them to be easily differentiated and integrated.
Unlike polynomials, \( y = \ln |x| \) is a logarithmic function. It doesn鈥檛 follow the standard polynomial form because it involves a logarithmic operation, showing that not all functions with basic elements like a single variable are polynomials. Instead, logarithmic functions have distinct properties, such as a domain defined by the input restrictions of the logarithm and potentially infinite ranges.
Unlike polynomials, \( y = \ln |x| \) is a logarithmic function. It doesn鈥檛 follow the standard polynomial form because it involves a logarithmic operation, showing that not all functions with basic elements like a single variable are polynomials. Instead, logarithmic functions have distinct properties, such as a domain defined by the input restrictions of the logarithm and potentially infinite ranges.
