Answer :
Let's analyze how the graph of function \(g(x) = -5 \ln x\) compares with the graph of function \(f(x) = \ln x\).
1. Vertical Asymptote:
Both functions \(f(x)\) and \(g(x)\) involve the natural logarithm \(\ln x\). The natural logarithm function \(\ln x\) has a vertical asymptote at \(x = 0\) because it approaches \(-\infty\) as \(x\) gets closer to 0 from the right. Since \(g(x)\) is simply a vertically scaled and reflected version of \(f(x)\), it also has a vertical asymptote at \(x = 0\).
Claim: True.
2. Domain:
The natural logarithm function \(f(x) = \ln x\) is defined only for \(x > 0\). Therefore, the domain of \(f(x)\) is \(\{x \mid x > 0\}\). Multiplying the logarithm by a constant (such as \(-5\) to get \(g(x) = -5 \ln x\)) does not alter the domain. Thus, the domain of \(g(x)\) is also \(\{x \mid x > 0\}\), not \(\{x \mid -5 < x < \infty\}\).
Claim: False.
3. Reflection and Vertical Stretch:
To obtain \(g(x)\) from \(f(x)\), we multiply \(f(x)\) by \(-5\). This reflects the graph of \(f(x)\) over the \(x\)-axis (because of the negative sign) and vertically stretches it by a factor of 5 (because of the coefficient 5).
Claim: True.
4. Behavior as \(x\) Increases:
The function \(f(x) = \ln x\) is an increasing function, meaning it increases as \(x\) increases. When you multiply by \(-5\), the function \(g(x) = -5 \ln x\) takes the increasing behavior of \(f(x)\) and turns it into decreasing behavior (the reflection effect). Thus, \(g(x)\) decreases as \(x\) increases.
Claim: True.
5. Y-intercept:
For a function to have a \(y\)-intercept, it must be defined at \(x = 0\). However, \(\ln x\) is not defined for \(x = 0\), and neither is \(-5 \ln x\). Therefore, neither function has a \(y\)-intercept.
Claim: False.
In summary, the correct claims are:
- The graphs of both functions have a vertical asymptote of \(x = 0\).
- The graph of function \(g\) is the graph of function \(f\) reflected over the \(x\)-axis and vertically stretched by a factor of 5.
- Unlike the graph of function [tex]\(f\)[/tex], the graph of function [tex]\(g\)[/tex] decreases as [tex]\(x\)[/tex] increases.
1. Vertical Asymptote:
Both functions \(f(x)\) and \(g(x)\) involve the natural logarithm \(\ln x\). The natural logarithm function \(\ln x\) has a vertical asymptote at \(x = 0\) because it approaches \(-\infty\) as \(x\) gets closer to 0 from the right. Since \(g(x)\) is simply a vertically scaled and reflected version of \(f(x)\), it also has a vertical asymptote at \(x = 0\).
Claim: True.
2. Domain:
The natural logarithm function \(f(x) = \ln x\) is defined only for \(x > 0\). Therefore, the domain of \(f(x)\) is \(\{x \mid x > 0\}\). Multiplying the logarithm by a constant (such as \(-5\) to get \(g(x) = -5 \ln x\)) does not alter the domain. Thus, the domain of \(g(x)\) is also \(\{x \mid x > 0\}\), not \(\{x \mid -5 < x < \infty\}\).
Claim: False.
3. Reflection and Vertical Stretch:
To obtain \(g(x)\) from \(f(x)\), we multiply \(f(x)\) by \(-5\). This reflects the graph of \(f(x)\) over the \(x\)-axis (because of the negative sign) and vertically stretches it by a factor of 5 (because of the coefficient 5).
Claim: True.
4. Behavior as \(x\) Increases:
The function \(f(x) = \ln x\) is an increasing function, meaning it increases as \(x\) increases. When you multiply by \(-5\), the function \(g(x) = -5 \ln x\) takes the increasing behavior of \(f(x)\) and turns it into decreasing behavior (the reflection effect). Thus, \(g(x)\) decreases as \(x\) increases.
Claim: True.
5. Y-intercept:
For a function to have a \(y\)-intercept, it must be defined at \(x = 0\). However, \(\ln x\) is not defined for \(x = 0\), and neither is \(-5 \ln x\). Therefore, neither function has a \(y\)-intercept.
Claim: False.
In summary, the correct claims are:
- The graphs of both functions have a vertical asymptote of \(x = 0\).
- The graph of function \(g\) is the graph of function \(f\) reflected over the \(x\)-axis and vertically stretched by a factor of 5.
- Unlike the graph of function [tex]\(f\)[/tex], the graph of function [tex]\(g\)[/tex] decreases as [tex]\(x\)[/tex] increases.
