Question

select all the correct answers.
consider function f and function g.
f(x) = ln x
g(x) = -5 ln x
how does the graph of function g compare with the graph of function f?
unlike the graph of function f, the graph of function g decreases as x increases.
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 f, the graph of function g has a domain of {x | - 5 < x < ∞}.
unlike the graph of function f, the graph of function g has a y - intercept.
the graphs of both functions have a vertical asymptote of x = 0.

Explanation:

Step1: Analyze the domain

The domain of \(y = \ln x\) is \(x>0\), and the domain of \(y=- 5\ln x\) is also \(x > 0\). So the statement about the domain of \(g\) is wrong.

Step2: Analyze the \(y -\)intercept

For \(y=\ln x\), when \(x = 1,y=0\). For \(y=-5\ln x\), when \(x = 1,y = 0\). The natural - logarithm function \(y=\ln x\) and \(y=-5\ln x\) have no \(y-\)intercept (because when \(x = 0\), \(\ln x\) is undefined). So the statement about the \(y-\)intercept is wrong.

Step3: Analyze the reflection and vertical stretch

The function \(g(x)=-5\ln x\) can be obtained from \(f(x)=\ln x\) by first reflecting \(y = \ln x\) over the \(x -\)axis (\(y=-\ln x\)) and then vertically stretching by a factor of \(5\) (\(y=-5\ln x\)).

Step4: Analyze the monotonicity

The derivative of \(f(x)=\ln x\) is \(f^\prime(x)=\frac{1}{x}>0\) for \(x>0\), so \(y = \ln x\) is increasing. The derivative of \(g(x)=-5\ln x\) is \(g^\prime(x)=-\frac{5}{x}<0\) for \(x > 0\), so \(y=-5\ln x\) is decreasing.

Step5: Analyze the vertical asymptote

For \(y=\ln x\), as \(x
ightarrow0^{+}\), \(y
ightarrow-\infty\). For \(y=-5\ln x\), as \(x
ightarrow0^{+}\), \(y
ightarrow+\infty\). Both \(y = \ln x\) and \(y=-5\ln x\) have a vertical asymptote \(x = 0\).

Answer:

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 \(f\), the graph of function \(g\) decreases as \(x\) increases; The graphs of both functions have a vertical asymptote of \(x = 0\).