To find the inverse of the function \( h(x) = -\frac{9x}{4} - \frac{1}{3} \), follow these steps:

### Steps:

1. **Rewrite the function:** Replace \( h(x) \) with \( y \):
   \[
   y = -\frac{9x}{4} - \frac{1}{3}
   \]

2. **Swap variables:** Switch \( x \) and \( y \) to prepare for solving the inverse:
   \[
   x = -\frac{9y}{4} - \frac{1}{3}
   \]

3. **Isolate \( y \):** Solve for \( y \) in terms of \( x \):

   - Add \( \frac{1}{3} \) to both sides:
     \[
     x + \frac{1}{3} = -\frac{9y}{4}
     \]

   - Multiply both sides by \( -\frac{4}{9} \) to isolate \( y \):
     \[
     y = -\frac{4}{9}\left(x + \frac{1}{3}\right)
     \]

4. **Simplify the expression:** Distribute \( -\frac{4}{9} \):
   \[
   y = -\frac{4}{9}x - \frac{4}{27}
   \]

Thus, the inverse function is:
\[
h^{-1}(x) = -\frac{4}{9}x - \frac{4}{27}
\]

### Do you have any questions, or would you like further details?

---

Here are 5 related questions to explore:

1. How do you verify if a function and its inverse are correct?
2. What are the conditions for a function to have an inverse?
3. How does the slope of a linear function relate to its inverse?
4. Can you describe the graph of the original function and its inverse?
5. How do inverse functions behave with respect to composition?

**Tip:** When solving for the inverse, always remember to switch \( x \) and \( y \) and then isolate \( y \).

Explore how to find the inverse of the linear function h(x) = -9x/4 - 1/3 with detailed steps. This problem covers essential algebraic concepts, including inverse functions and linear equations. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation