Question

y = -lnx y = ln(-x) y = -ln(-x) y = lnx

Explanation:

Step1: Analyze the domain of each function

• For \(y = \ln x\), the domain is \(x>0\).
• For \(y =-\ln x\), the domain is \(x > 0\).
• For \(y=\ln(-x)\), the domain is \(x<0\).
• For \(y =-\ln(-x)\), the domain is \(x<0\).

Step2: Analyze the behavior of \(y = \ln x\)

The function \(y=\ln x\) has a vertical asymptote at \(x = 0\), passes through the point \((1,0)\), and is an increasing function for \(x>0\).

Step3: Analyze the behavior of \(y=-\ln x\)

Since \(y =-\ln x=-(\ln x)\), it is the reflection of \(y = \ln x\) about the \(x\) - axis. It has a vertical asymptote at \(x = 0\), passes through the point \((1,0)\), and is a decreasing function for \(x>0\).

Step4: Analyze the behavior of \(y=\ln(-x)\)

Let \(t=-x\). When \(x<0\), \(t>0\). The function \(y = \ln(-x)\) is the reflection of \(y=\ln x\) about the \(y\) - axis. It has a vertical asymptote at \(x = 0\), passes through the point \((- 1,0)\), and is an increasing function for \(x<0\).

Step5: Analyze the behavior of \(y=-\ln(-x)\)

Since \(y=-\ln(-x)=-(\ln(-x))\), it is the reflection of \(y = \ln(-x)\) about the \(x\) - axis. It has a vertical asymptote at \(x = 0\), passes through the point \((-1,0)\), and is a decreasing function for \(x<0\).

Step6: Match the graphs

• The first graph (left - top) with domain \(x>0\) and increasing: \(y=\ln x\).
• The second graph (right - top) with domain \(x>0\) and decreasing: \(y =-\ln x\).
• The third graph (left - bottom) with domain \(x<0\) and increasing: \(y=\ln(-x)\).
• The fourth graph (right - bottom) with domain \(x<0\) and decreasing: \(y=-\ln(-x)\).

Answer:

First graph (left - top): \(y = \ln x\); Second graph (right - top): \(y=-\ln x\); Third graph (left - bottom): \(y=\ln(-x)\); Fourth graph (right - bottom): \(y =-\ln(-x)\)