Problem 211
Question
For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they do not exist, write DNE. $$g(x)=\ln (-x)-2$$
Step-by-Step Solution
Verified Answer
Domain: \((-\infty, 0)\); Range: \((-\infty, \infty)\); \(x\)-intercept: \((-e^2, 0)\); \(y\)-intercept: DNE.
1Step 1: Determine the Domain
The domain of a logarithmic function is determined by ensuring the argument of the logarithm is greater than zero. For the function \( g(x) = \ln(-x) - 2 \), the argument is \(-x\). Set it greater than zero: \(-x > 0\). Solving this inequality, we get \(x < 0\). Therefore, the domain of \( g(x) \) is all negative real numbers: \((-\infty, 0)\).
2Step 2: Determine the Range
The range of a logarithmic function, such as \( \ln(x) \), is all real numbers. Since \( g(x) = \ln(-x) - 2 \) involves just a vertical shift of \( \ln(-x) \) downward by 2 units, the range remains as all real numbers: \((-\infty, \infty)\).
3Step 3: Find the x-intercept
To find the \(x\)-intercept, set \(g(x) = 0\):\( \ln(-x) - 2 = 0 \).Add 2 to both sides:\( \ln(-x) = 2 \).Exponentiate both sides:\( e^{\ln(-x)} = e^2 \).Which gives \(-x = e^2\) or \(x = -e^2\). So, the \(x\)-intercept is \((-e^2, 0)\).
4Step 4: Find the y-intercept
To find the \(y\)-intercept, set \(x = 0\) in the function:\( g(0) = \ln(0) - 2 \).However, \( \ln(0) \) is undefined. Additionally, since \(x = 0\) is not in the domain \((-\infty, 0)\), the function does not have a \(y\)-intercept. Therefore, the \(y\)-intercept is DNE (Does Not Exist).
Key Concepts
Understanding Domain and RangeFinding the Efficient X-InterceptUnderstanding the Y-Intercept DilemmaHandling Functional Inequalities
Understanding Domain and Range
In mathematics, understanding the domain and range of functions is essential. The domain of a function defines all the possible input values (x-values) for which the function is defined. For logarithmic functions, specifically, ensuring that the argument inside the logarithm function is greater than zero is crucial. For the function \( g(x) = \ln(-x) - 2 \), the argument \(-x\) must be positive, meaning \(-x > 0\). This simplifies to \(x < 0\). Consequently, the domain of this function encompasses all negative real numbers, which can be expressed as the interval \(( -\infty, 0)\).
The range of a function, on the other hand, describes all the possible output values (y-values). For a standard logarithmic function like \( \ln(x) \), the range is all real numbers, \(( -\infty, \infty)\). Since \( g(x) \) only involves a vertical shift, moving the entire graph downwards by 2 units, its range remains unchanged. This means the range is still all real numbers, from \(-\infty\) to \(\infty\). Understanding these concepts ensures you know the behavior and limitations of the function.
The range of a function, on the other hand, describes all the possible output values (y-values). For a standard logarithmic function like \( \ln(x) \), the range is all real numbers, \(( -\infty, \infty)\). Since \( g(x) \) only involves a vertical shift, moving the entire graph downwards by 2 units, its range remains unchanged. This means the range is still all real numbers, from \(-\infty\) to \(\infty\). Understanding these concepts ensures you know the behavior and limitations of the function.
Finding the Efficient X-Intercept
Determining where a function crosses the x-axis is essential for graphing and understanding the function's behavior. This point is referred to as the x-intercept. To find it for \( g(x) = \ln(-x) - 2 \), we set the function equal to zero: \( \ln(-x) - 2 = 0 \). Solving involves isolating the logarithmic part, which leads to \( \ln(-x) = 2 \).
To eliminate the logarithm and simplify, exponentiate both sides, resulting in \( e^{\ln(-x)} = e^2 \). The exponential function and the natural logarithm are inverse operations, hence simplifying to \(-x = e^2\). Solving for \(x\) yields \( x = -e^2 \). Therefore, the x-intercept occurs at \((-e^2, 0)\).
This point indicates where the function touches the x-axis, which is an important aspect of graph determination and functional analysis.
To eliminate the logarithm and simplify, exponentiate both sides, resulting in \( e^{\ln(-x)} = e^2 \). The exponential function and the natural logarithm are inverse operations, hence simplifying to \(-x = e^2\). Solving for \(x\) yields \( x = -e^2 \). Therefore, the x-intercept occurs at \((-e^2, 0)\).
This point indicates where the function touches the x-axis, which is an important aspect of graph determination and functional analysis.
Understanding the Y-Intercept Dilemma
The y-intercept is the point where a function crosses the y-axis, found by setting \( x = 0 \) in the equation and solving for \( y \). However, for \( g(x) = \ln(-x) - 2 \), when \(x = 0\), you attempt \( g(0) = \ln(0) - 2 \).
The problem here is mathematical as \( \ln(0) \) is undefined; you cannot take the natural logarithm of zero in the real number system. Moreover, the domain restriction \(( -\infty, 0)\) excludes zero. This combination firmly denotes that the y-intercept for this function does not exist, abbreviated as DNE.
Recognizing when intercepts do not exist is just as valuable as finding them. It reveals information about the function's graph and its domain constraints.
The problem here is mathematical as \( \ln(0) \) is undefined; you cannot take the natural logarithm of zero in the real number system. Moreover, the domain restriction \(( -\infty, 0)\) excludes zero. This combination firmly denotes that the y-intercept for this function does not exist, abbreviated as DNE.
Recognizing when intercepts do not exist is just as valuable as finding them. It reveals information about the function's graph and its domain constraints.
Handling Functional Inequalities
Inequalities in functions like \( g(x) = \ln(-x) - 2 \) necessitate understanding domain restrictions to resolve mathematical operations. When solving inequalities, always focus firstly on domains. Here, because of \( \ln(-x) \), \(-x > 0\) translates to \(x < 0\). This distinction prevents errors when considering possible input values.
For logarithmic inequalities, especially, it is crucial to remember that the inequality sign flips when multiplying or dividing by a negative number. Through such careful handling of inequalities, you ensure valid and correct results considering the function's nature. This approach aids in clearly understanding restrictions and how inequalities guide the solutions.
For logarithmic inequalities, especially, it is crucial to remember that the inequality sign flips when multiplying or dividing by a negative number. Through such careful handling of inequalities, you ensure valid and correct results considering the function's nature. This approach aids in clearly understanding restrictions and how inequalities guide the solutions.
Other exercises in this chapter
Problem 209
For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they do not exist, write DNE. $$h(x)=\log _{4}(x-1)+1$$
View solution Problem 210
For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they do not exist, write DNE. $$f(x)=\log (5 x+10)+3$$
View solution Problem 212
For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they do not exist, write DNE. $$f(x)=\log _{2}(x+2)-5$$
View solution Problem 213
For the following exercises, state the domain, range, and \(x\) - and \(y\) -intercepts, if they do not exist, write DNE. $$h(x)=3 \ln (x)-9$$
View solution