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To determine whether the two functions \( f(n) = 2 - \frac{3}{2}n \) and \( h(n) = -\frac{2}{3}n + \frac{4}{3} \) are inverses of each other, we need to check if applying one function to the output of the other results in \( n \), the input.

In other words, we need to confirm whether:
- \( f(h(n)) = n \)
- \( h(f(n)) = n \)

Let's compute both compositions step by step:

### 1. Finding \( f(h(n)) \):
Substitute \( h(n) \) into \( f(n) \):
\[
f(h(n)) = 2 - \frac{3}{2} \left(-\frac{2}{3}n + \frac{4}{3}\right)
\]
Simplify this expression:
\[
f(h(n)) = 2 - \frac{3}{2} \cdot \left(-\frac{2}{3}n\right) - \frac{3}{2} \cdot \frac{4}{3}
\]
\[
f(h(n)) = 2 + n - 2
\]
\[
f(h(n)) = n
\]

### 2. Finding \( h(f(n)) \):
Substitute \( f(n) \) into \( h(n) \):
\[
h(f(n)) = -\frac{2}{3} \left(2 - \frac{3}{2}n\right) + \frac{4}{3}
\]
Simplify this expression:
\[
h(f(n)) = -\frac{2}{3} \cdot 2 + \frac{2}{3} \cdot \frac{3}{2}n + \frac{4}{3}
\]
\[
h(f(n)) = -\frac{4}{3} + n + \frac{4}{3}
\]
\[
h(f(n)) = n
\]

Since both \( f(h(n)) = n \) and \( h(f(n)) = n \), the two functions are indeed inverses of each other.

### Conclusion:
The correct answer is **B) Yes**.

Would you like more details or have any questions?

### Related Questions:
1. What are the steps to find the inverse of a function algebraically?
2. Can every function have an inverse? Why or why not?
3. How can you determine if a function is one-to-one and thus has an inverse?
4. What is the geometric relationship between a function and its inverse on a graph?
5. How do inverse functions relate to real-world applications, such as conversions between units?

**Tip**: To check if two functions are inverses, always check both compositions: \( f(h(n)) \) and \( h(f(n)) \). Both must simplify to \( n \).

State if the given functions are inverses. f(n) = 2 - (3/2)n, h(n) = -(2/3)n + 4/3

Learn how to check if two functions f(n) = 2 - (3/2)n and h(n) = -(2/3)n + 4/3 are inverses of each other using function composition. Suitable for students in Grades 9-11.

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