select all the correct answers. consider function f and function g. f(x) = ln x g(x) = -5 ln x how does the…

Explanation:

Step1: Analyze the derivative of the functions

The derivative of $f(x)=\ln x$ is $f^\prime(x)=\frac{1}{x}$, and for $g(x)= - 5\ln x$, its derivative is $g^\prime(x)=-\frac{5}{x}$. For $x>0$, when $x$ increases, $f^\prime(x)>0$ so $f(x)$ increases, and $g^\prime(x)<0$ so $g(x)$ decreases.

Step2: Check the y - intercept

The domain of $y = \ln x$ and $y=-5\ln x$ is $x>0$. Since $x = 0$ is not in the domain, neither function has a y - intercept.

Step3: Determine the vertical asymptote

As $x\rightarrow0^{+}$, $\ln x\rightarrow-\infty$ and $-5\ln x\rightarrow\infty$. So the vertical asymptote of both $y = f(x)$ and $y = g(x)$ is $x = 0$.

Step4: Analyze the transformation

The function $g(x)=-5\ln x=- 5\times f(x)$. The negative sign reflects the graph of $f(x)$ over the x - axis, and the coefficient 5 vertically stretches the graph by a factor of 5.

Step5: Check the domain

The domain of $f(x)=\ln x$ and $g(x)=-5\ln x$ is $x>0$, not ${x|-5 < x<\infty}$.

Answer:

• A. Unlike the graph of function $f$, the graph of function $g$ decreases as $x$ increases.
• C. The graphs of both functions have a vertical asymptote of $x = 0$.
• D. The graph of function $g$ is the graph of function $f$ reflected over the x - axis and vertically stretched by a factor of 5.