State the domain, vertical asymptote, and end behavior of the function.

 

h(x)=−log(3x−8)+6

Let's analyze the function \( h(x) = -\log(3x - 8) + 6 \) to find its **domain**, **vertical asymptote**, and **end behavior**.

### 1. Domain:
The function involves a logarithmic expression \( \log(3x - 8) \), and the argument of the logarithm must be strictly positive, meaning:

\[
3x - 8 > 0
\]

Solve for \( x \):

\[
3x > 8 \quad \Rightarrow \quad x > \frac{8}{3}
\]

Thus, the domain of \( h(x) \) is:

\[
x > \frac{8}{3}
\]

or in interval notation:

\[
\boxed{ \left( \frac{8}{3}, \infty \right) }
\]

### 2. Vertical Asymptote:
A vertical asymptote occurs where the argument of the logarithm approaches zero, which happens when \( 3x - 8 = 0 \). Solve for \( x \):

\[
3x = 8 \quad \Rightarrow \quad x = \frac{8}{3}
\]

Thus, the vertical asymptote is:

\[
\boxed{ x = \frac{8}{3} }
\]

### 3. End Behavior:
To analyze the end behavior, we examine how \( h(x) \) behaves as \( x \to \infty \).

- As \( x \to \infty \), \( 3x - 8 \to \infty \), so \( \log(3x - 8) \to \infty \).
- Since \( h(x) = -\log(3x - 8) + 6 \), the logarithmic part grows large, but the negative sign flips the behavior. Hence, \( -\log(3x - 8) \to -\infty \).
- Adding 6 gives \( h(x) \to -\infty \) as \( x \to \infty \).

So, the end behavior is:

\[
\boxed{ \lim_{x \to \infty} h(x) = -\infty }
\]

### Summary:
- **Domain**: \( \left( \frac{8}{3}, \infty \right) \)
- **Vertical Asymptote**: \( x = \frac{8}{3} \)
- **End Behavior**: As \( x \to \infty \), \( h(x) \to -\infty \).

Do you need further clarification or details? 

Here are 5 related questions to deepen understanding:
1. How does the vertical asymptote affect the graph of a logarithmic function?
2. What happens to the graph of \( h(x) \) as \( x \) approaches the vertical asymptote from the right?
3. How would the domain change if the logarithmic expression had a different base?
4. How does adding or subtracting constants outside the logarithm affect the graph of the function?
5. What if the argument of the logarithmic function had a negative coefficient? How would that impact the graph?

**Tip**: Always check the argument of a logarithmic function to ensure it remains positive to determine the domain correctly.

State the domain, vertical asymptote, and end behavior of the function h(x) = −log(3x−8) + 6.

Analyze the domain, vertical asymptote, and end behavior of the function h(x) = −log(3x−8) + 6. This problem covers logarithmic functions, asymptotes, and function behavior as x approaches infinity. Suitable for high school students in Grades 11-12.

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