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Magazine Chapter 5: Problem 55
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
Applicable properties: B, D, E.
Step by step solution
01
Understand the Function
The given function is \(y = -\ln x\), which is the negative logarithm of \(x\). This is a logarithmic function, not a polynomial.
02
Determine Increasing or Decreasing
The natural logarithm function \(\ln x\) is increasing for all \(x > 0\). Therefore, \(-\ln x\) will be decreasing for \(x > 0\). So, the function decreases wherever it is defined.
03
Assess Turns or Asymptotes
The function \(-\ln x\) does not have any turning points, as it consistently decreases. It does have an asymptote at \(x = 0\) since the logarithmic function is undefined at and approaching \(x = 0\) from the right.
04
Verify One-to-One Property
A one-to-one function is a function that never assigns the same value to two different domain elements. Since \(-\ln x\) is a monotonically decreasing function when \(x > 0\), it never gives the same output for different inputs, thus it is one-to-one.
05
Identify the Domain and Range
The domain of \(y=-\ln x\) is \((0, \infty)\) because the logarithm is only defined for \(x > 0\). The range is \(( -\infty, \infty )\) as \(y\) values range from positive infinity to negative infinity as \(x\) approaches zero from the right and becomes very large.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Functions
Logarithmic functions are a special type of mathematical function and are the inverse of exponential functions. They are often written in the form of \(y = \ln x\), where \(\ln\) stands for natural logarithm, which means log base \(e\), with \(e\) being approximately 2.718. In this case, we look at \(y = -\ln x\), the negative natural logarithm of \(x\).
- Logarithmic functions grow or shrink very slowly compared to polynomial or exponential functions.
- They are only defined for positive values of \(x\) because you can't take the log of a negative number or zero.
Asymptotes
An important property of logarithmic functions is that they often have asymptotes. An asymptote is a line that the graph of a function approaches but never actually touches. For logarithmic functions like \(-\ln x\), an asymptote occurs at \(x = 0\).
- As \(x\) gets closer to zero from the right, \(-\ln x\) decreases without bound, meaning the function heads towards negative infinity.
- The vertical line \(x = 0\) is never a part of the graph but acts as a boundary.
Increasing and Decreasing Functions
Functions can be classified as increasing or decreasing based on how they behave over their domains. For the function \(y = -\ln x\):
- The natural logarithm \(\ln x\) is an increasing function for \(x > 0\).
- By taking the negative, \(-\ln x\) simply reverses this increase to a decrease.
- Hence, \(y = -\ln x\) is decreasing for all \(x > 0\).
Domain and Range
The domain and range are essential to understanding any function, as they denote what inputs \(x\) are possible and what outputs \(y\) are likely.
- For \(y = -\ln x\), the domain is \((0, \infty)\). This means \(x\) must be positive because logarithms are not defined for zero or negative values.
- The range of this function is \(( -\infty, \infty )\). As \(x\) approaches infinity, \(y\) approaches negative values, and as \(x\) nears zero from the right, \(y\) goes to positive infinity.
