state the domain and the vertical asymptote of the function: g(x)=ln(4-x). enter the domain in interval notation$$  $$

To determine the domain and vertical asymptote of the function $ g(x) = \ln(4 - x) $, let's break it down:

### Domain:
The natural logarithm function, $ \ln(x) $, is only defined for positive values of its argument. In this case, the argument of the logarithm is $ 4 - x $. Thus, for the function to be defined:

\[
4 - x > 0
\]

Solving this inequality:

\[
x < 4
\]

So, the domain is all values of $ x $ less than 4. In interval notation, the domain is:

\[
(-\infty, 4)
\]

### Vertical Asymptote:
A vertical asymptote occurs when the function approaches infinity as $ x $ approaches a certain value. For logarithmic functions, the vertical asymptote occurs where the argument of the logarithm is 0.

Set the argument of the logarithm equal to 0:

\[
4 - x = 0 \quad \Rightarrow \quad x = 4
\]

Thus, there is a vertical asymptote at $ x = 4 $.

### Summary:
- **Domain:** $ (-\infty, 4) $
- **Vertical Asymptote:** $ x = 4 $

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the range of the function $ g(x) = \ln(4 - x) $?
2. How does the graph of $ g(x) = \ln(4 - x) $ compare to $ f(x) = \ln(x) $?
3. What is the derivative of $ g(x) = \ln(4 - x) $?
4. Where does the function $ g(x) = \ln(4 - x) $ have horizontal asymptotes, if any?
5. Find the inverse of the function $ g(x) = \ln(4 - x) $.

**Tip:** When finding the domain of a logarithmic function, always ensure the argument inside the logarithm is positive.

state the domain and the vertical asymptote of the function: g(x)=ln(4-x). enter the domain in interval notation

Learn how to find the domain and vertical asymptote of the logarithmic function g(x) = ln(4 - x). This problem explores logarithmic functions, inequalities, and vertical asymptotes, suitable for students in Grades 10-12.

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