Question

score: 4/15 answered: 4/15 question 5 find the vertical asymptote, domain, and range of the following exponential function. f(x) = ln(x - 2) - 4 a) find the vertical asymptote. be sure to give an equation. b) state the domain using interval notation. c) state the range using interval notation. question help: video 1 video 2 submit question

Explanation:

Part (a)

Step 1: Recall the vertical asymptote of the natural logarithm function

The parent function \( y = \ln(x) \) has a vertical asymptote at \( x = 0 \) because the natural logarithm is undefined at \( x = 0 \) and approaches negative infinity as \( x \) approaches \( 0 \) from the right. For the function \( f(x)=\ln(x - 2)-4 \), we need to find where the argument of the logarithm is zero, since the transformation (subtracting 2 inside the log and subtracting 4 outside) does not affect the vertical asymptote's position (horizontal shifts affect the vertical asymptote, vertical shifts do not).

Set the argument of the logarithm equal to zero: \( x - 2=0 \)

Step 2: Solve for \( x \)

Solving \( x - 2 = 0 \) gives \( x=2 \). So the vertical asymptote of \( f(x)=\ln(x - 2)-4 \) is \( x = 2 \).

Part (b)

Step 1: Recall the domain of the natural logarithm function

The domain of \( y=\ln(x) \) is \( x>0 \) (all positive real numbers). For the function \( f(x)=\ln(x - 2)-4 \), the argument of the logarithm \( x - 2 \) must be greater than zero.

Step 2: Solve the inequality for \( x \)

Set up the inequality: \( x - 2>0 \)
Add 2 to both sides: \( x>2 \)
In interval notation, this is \( (2, \infty) \).

Part (c)

Step 1: Recall the range of the natural logarithm function

The range of \( y = \ln(x) \) is all real numbers, \( (-\infty, \infty) \). A vertical shift (subtracting 4) does not change the range of a function, it only shifts the graph up or down. So the range of \( f(x)=\ln(x - 2)-4 \) is also all real numbers.

Step 2: Write the range in interval notation

The interval notation for all real numbers is \( (-\infty, \infty) \).

Answer:

s:
a) The vertical asymptote is \( \boldsymbol{x = 2} \)
b) The domain in interval notation is \( \boldsymbol{(2, \infty)} \)
c) The range in interval notation is \( \boldsymbol{(-\infty, \infty)} \)