https://file.aiquickdraw.com/mathbot/file/1729765870665f4qngc2d.jpg

The given function is:

\[
f(x) = -\log_2(3x)
\]

This is a logarithmic function with a base of 2, and the argument inside the logarithm is \( 3x \). The negative sign in front of the logarithm means that the values of the function will be negated.

### Key Properties:

1. **Domain**: Since the argument of a logarithmic function must be greater than 0, we have:
   \[
   3x > 0 \implies x > 0
   \]
   So, the domain of the function is \( (0, \infty) \).

2. **Range**: The logarithmic function \( \log_2(3x) \) has a range of \( (-\infty, \infty) \). However, due to the negative sign, the function \( f(x) \) will output values in \( (-\infty, \infty) \).

3. **Asymptote**: There is a vertical asymptote at \( x = 0 \), as \( \log_2(3x) \to -\infty \) when \( x \to 0^+ \).

4. **Behavior**:
   - As \( x \to 0^+ \), \( f(x) \to \infty \) because of the negative logarithm.
   - As \( x \to \infty \), \( f(x) \to -\infty \).

Would you like to go into further details about how to solve or analyze this function?

Here are 5 questions related to this topic:

1. How would you find the inverse of the function \( f(x) = -\log_2(3x) \)?
2. What is the derivative of the function \( f(x) \)?
3. Can you graph \( f(x) = -\log_2(3x) \) and explain the key features?
4. How do you solve \( f(x) = 0 \) for \( x \)?
5. What happens to the function if you change the base of the logarithm?

**Tip:** Logarithmic transformations can help linearize exponential data, making it easier to analyze complex relationships.

Explore the properties of the logarithmic function f(x) = -log_2(3x). Learn about its domain, range, asymptotes, and behavior as x approaches different limits. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation