Questions: Begin by graphing (g(x)=ln x) Use transformations of this graph to graph the given function. Graph and give the equation of the asymptote. Use the graphs to determine the function's domain and range. [g(x)=ln (x+6)] Graph (g(x)=ln (x+6)). Graph the asymptote of (g(x)) as a dashed line. Use the graphing tool to graph the function. What is the vertical asymptote of (g(x)) ? (Type an equation) What is the domain of (g(x)=ln (x+6)) ? (Simplify your answer. Type your answer in interval notation.) What is the range of (g(x)=ln (x+6)) ? (Simplify your answer. Type your answer in interval notation.)

Begin by graphing (g(x)=ln x) Use transformations of this graph to graph the given function. Graph and give the equation of the asymptote. Use the graphs to determine the function's domain and range. [g(x)=ln (x+6)] Graph (g(x)=ln (x+6)). Graph the asymptote of (g(x)) as a dashed line. Use the graphing tool to graph the function. What is the vertical asymptote of (g(x)) ? (Type an equation) What is the domain of (g(x)=ln (x+6)) ? (Simplify your answer. Type your answer in interval notation.) What is the range of (g(x)=ln (x+6)) ? (Simplify your answer. Type your answer in interval notation.)
Transcript text: Begin by graphing $(x)=\ln x$ Use transformations of this graph to graph the given function. Graph and give the equation of the asymptote. Use the graphs to determine the function's domain and range. \[ g(x)=\ln (x+6) \] Graph $g(x)=\ln (x+6)$. Graph the asymptote of $g(x)$ as a dashed line. Use the graphing tool to graph the function. What is the vertical asymptote of $g(x)$ ? $\square$ (Type an equation) What is the domain of $g(x)=\ln (x+6)$ ? $\square$ (Simplify your answer. Type your answer in interval notation.) What is the range of $g(x)=\ln (x+6)$ ? $\square$ (Simplify your answer. Type your answer in interval notation.)
failed

Solution

failed
failed

Solution Steps

Step 1: Determine the vertical asymptote

The vertical asymptote of the function \( g(x) = \ln(x+6) \) occurs where the argument of the logarithm is zero: \[ x + 6 = 0 \implies x = -6 \]

Step 2: Determine the domain of the function

The domain of \( g(x) = \ln(x+6) \) is where the argument of the logarithm is positive: \[ x + 6 > 0 \implies x > -6 \] Thus, the domain in interval notation is: \[ (-6, \infty) \]

Step 3: Determine the range of the function

The range of the natural logarithm function is all real numbers: \[ (-\infty, \infty) \]

Final Answer

  • The vertical asymptote of \( g(x) = \ln(x+6) \) is \( x = -6 \).
  • The domain of \( g(x) = \ln(x+6) \) is \( (-6, \infty) \).
  • The range of \( g(x) = \ln(x+6) \) is \( (-\infty, \infty) \).

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = ln(x + 6)", "x = -6"], "latex_expressions": ["$y = \\ln(x + 6)$", "$x = -6$"]}

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
The data given below show the number of overtime hours worked in one week per employee. Use the data to complete parts (a) and (b). Overtime hours 0 1 2 3 4 5 6 b 6 12 27 59 46 28 19 Employees (a) Construct a probability distribution. (b) Graph the probability distribution using a histogram and describe its shape. (a) Construct the probability distribution by completing the table below x 0 P(x) 1 2 3 4 5 6 P(x) (Round to three decimal places as needed.)
failedAnswered
A rectangular room is 2 times as long as it is wide, and its perimeter is 48 meters. Find the dimension of the room. The length is : meters and the width is meters.
failedAnswered
x(t) = 2 + 3/t, y(t) = t-1, 2 ≤ t ≤ 6
failedAnswered
How did the Columbian exchange impact Europeans and Native Americans?
failedAnswered
Factor. 14x^2+27x-20