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.
• unlike the graph of function $f$, the graph of function $g$ has a domain of $\\{x|-5 < x < \infty\\}$
• the graphs of both functions have a vertical asymptote of $x = 0$.
• unlike the graph of function $f$, the graph of function $g$ has a $y-$intercept.
• the graph of function $g$ is the graph of function $f$ reflected over the x-axis and vertically stretched by a factor of 5.

Explanation:

Step1: Analyze monotonicity

$f(x)=\ln x$ increases as $x$ increases. $g(x)=-5\ln x$, its derivative $g'(x)=-\frac{5}{x}<0$ for $x>0$, so $g(x)$ decreases as $x$ increases.

Step2: Analyze domain

Domain of $f(x)=\ln x$ is $\{x|x>0\}$. Domain of $g(x)=-5\ln x$ is also $\{x|x>0\}$, since $\ln x$ requires $x>0$.

Step3: Analyze vertical asymptote

For $f(x)=\ln x$, $\lim_{x\to0^+}\ln x=-\infty$, so vertical asymptote $x=0$. For $g(x)=-5\ln x$, $\lim_{x\to0^+}-5\ln x=+\infty$, so vertical asymptote $x=0$.

Step4: Analyze y-intercept

Y-intercept occurs at $x=0$, but $f(x)$ and $g(x)$ are undefined at $x=0$, so neither has a y-intercept.

Step5: Analyze transformation

For function $y=k\cdot f(x)$, when $k<0$, it is reflection over x-axis and vertical stretch by $|k|$. Here $g(x)=-5f(x)$, so it is reflection of $f(x)$ over x-axis and vertical stretch by factor 5.

Answer:

• 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 $x = 0$.
• The graph of function $g$ is the graph of function $f$ reflected over the x-axis and vertically stretched by a factor of 5.