answer the questions below about the following function:
$f(x) = -4 \ln(x)$
(there is a graph of the function here)
\begin{itemize}
\itema) what is the domain of $f(x)$?
\itemb) what is the range of $f(x)$?
\itemc) what is the $x$-intercept of $f(x)$? enter your answer as an ordered pair. if no $x$-intercept exists, enter none.
\itemd) is $f(x)$ an increasing or decreasing function?
\begin{itemize}
\item increasing
\item decreasing
\end{itemize}
\iteme) what is the equation of the vertical asymptote?
\end{itemize}

Question

Accepted Answer

### Part (a) # Explanation: ## Step1: Recall domain of logarithm The function is $ f(x) = -4\ln(x) $. The natural logarithm $ \ln(x) $ is defined only when $ x > 0 $ (since the argument of a logarithm must be positive). ## Step2: Determine domain of $ f(x) $ Since the only restriction comes from the $ \ln(x) $ term, the domain of $ f(x) = -4\ln(x) $ is all real numbers $ x $ such that $ x > 0 $. In interval notation, this is $ (0, \infty) $. # Answer: The domain of $ f(x) $ is $ (0, \infty) $ (or all real numbers $ x $ where $ x > 0 $) ### Part (b) # Explanation: ## Step1: Analyze the range of $ \ln(x) $ The natural logarithm function $ \ln(x) $ has a range of $ (-\infty, \infty) $ (all real numbers) because as $ x $ approaches $ 0^+ $, $ \ln(x) $ approaches $ -\infty $, and as $ x $ approaches $ \infty $, $ \ln(x) $ approaches $ \infty $. ## Step2: Determine the range of $ -4\ln(x) $ Multiplying $ \ln(x) $ by $ -4 $ (a constant) does not change the range in terms of being all real numbers. If $ \ln(x) $ can take any real value, then $ -4\ln(x) $ can also take any real value (since multiplying by a non - zero constant just scales the values, not restricts the set of possible values). So the range of $ f(x)=-4\ln(x) $ is $ (-\infty, \infty) $ (or all real numbers). # Answer: The range of $ f(x) $ is $ (-\infty, \infty) $ (or all real numbers) ### Part (c) # Explanation: ## Step1: Recall x - intercept definition The x - intercept of a function $ y = f(x) $ is the point where $ y = 0 $, i.e., we need to solve $ f(x)=0 $ for $ x $. ## Step2: Solve $ -4\ln(x)=0 $ Divide both sides of the equation $ -4\ln(x)=0 $ by $ -4 $. We get $ \ln(x)=0 $. We know that the natural logarithm function $ \ln(x) $ is the inverse of the exponential function $ e^x $. So, if $ \ln(x)=0 $, then by the definition of logarithms, $ x = e^0 $. Since $ e^0 = 1 $, when $ x = 1 $, $ f(1)=-4\ln(1)=-4\times0 = 0 $. So the x - intercept is the ordered pair $ (1, 0) $. # Answer: The x - intercept of $ f(x) $ is $ (1, 0) $ ### Part (d) # Explanation: ## Step1: Recall the derivative of $ \ln(x) $ The derivative of $ \ln(x) $ with respect to $ x $ is $ \frac{1}{x} $. For the function $ f(x)=-4\ln(x) $, we use the constant multiple rule of differentiation. The derivative $ f^\prime(x)=-4\times\frac{1}{x}=-\frac{4}{x} $. ## Step2: Analyze the sign of the derivative For the domain of $ f(x) $ (where $ x>0 $), $ \frac{4}{x}>0 $, so $ -\frac{4}{x}<0 $ for all $ x $ in the domain $ (0, \infty) $. A function is decreasing on an interval if its derivative is negative on that interval. Since $ f^\prime(x)=-\frac{4}{x}<0 $ for all $ x > 0 $, the function $ f(x) $ is a decreasing function. # Answer: The function $ f(x) $ is a decreasing function. So the answer is Decreasing. ### Part (e) # Explanation: ## Step1: Recall vertical asymptote of $ \ln(x) $ The function $ y = \ln(x) $ has a vertical asymptote at $ x = 0 $ (as $ x $ approaches $ 0^+ $, $ \ln(x) $ approaches $ -\infty $). ## Step2: Determine vertical asymptote of $ -4\ln(x) $ Multiplying $ \ln(x) $ by $ -4 $ does not change the location of the vertical asymptote. The behavior of $ f(x)=-4\ln(x) $ as $ x $ approaches $ 0^+ $ is that $ \ln(x) $ approaches $ -\infty $, so $ -4\ln(x) $ approaches $ \infty $, and as $ x $ approaches $ 0^+ $, the function values shoot off to infinity (in this case, positive infinity). So the vertical asymptote is at $ x = 0 $. # Answer: The equation of the vertical asymptote is $ x = 0 $

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Recall domain of logarithm

Step2: Determine domain of \( f(x) \)

Step1: Analyze the range of \( \ln(x) \)

Step2: Determine the range of \( -4\ln(x) \)

Step1: Recall x - intercept definition

Step2: Solve \( -4\ln(x)=0 \)

Answer:

Part (b)