- For Part 5, the composition \(f(g(n)) = \frac{3n-10}{8}\), which is not equal to \(n\), so they are not inverses.
- For Part 6, the compositions \(f(g(n)) = n\) and \(g(f(n)) = n\), confirming the functions are inverses.
- The results are: 5) No, 6) Yes. \(\boxed{\text{5) No, 6) Yes}}\)
Explanation
- Defining Inverse Functions
To determine if two functions \(f(n)\) and \(g(n)\) are inverses of each other, we must verify if the composition of the two functions returns the input variable \(n\). Specifically, \(f\) and \(g\) are inverses if and only if \(f(g(n)) = n\) and \(g(f(n)) = n\) for all \(n\) in the domain. We will check this condition for both given pairs.
- Evaluating Part 5
For Part 5, we have \(g(n) = \frac{-n + 6}{4}\) and \(f(n) = 1 - \frac{3}{2}n\). We evaluate the composition \(f(g(n))\):
\[f(g(n)) = f\left(\frac{-n + 6}{4}\right) = 1 - \frac{3}{2} \left( \frac{-n + 6}{4} \right)\]
Multiplying the terms:
\[f(g(n)) = 1 - \frac{3(-n + 6)}{8} = 1 - \left( \frac{-3n + 18}{8} \right)\]
Combining over a common denominator:
\[f(g(n)) = \frac{8}{8} - \frac{-3n + 18}{8} = \frac{8 + 3n - 18}{8} = \frac{3n - 10}{8}\]
Since \(\frac{3n - 10}{8}
eq n\), the functions are not inverses.
- Evaluating Part 6
For Part 6, we have \(g(n) = \frac{2 + 5n}{2}\) and \(f(n) = \frac{2n - 2}{5}\). We evaluate the compositions \(f(g(n))\) and \(g(f(n))\):
First, \(f(g(n))\):
\[f(g(n)) = f\left(\frac{2 + 5n}{2}\right) = \frac{2\left(\frac{2 + 5n}{2}\right) - 2}{5} = \frac{(2 + 5n) - 2}{5} = \frac{5n}{5} = n\]
Next, \(g(f(n))\):
\[g(f(n)) = g\left(\frac{2n - 2}{5}\right) = \frac{2 + 5\left(\frac{2n - 2}{5}\right)}{2} = \frac{2 + (2n - 2)}{2} = \frac{2n}{2} = n\]
Since both compositions result in \(n\), the functions in Part 6 are inverses.
Examples
Inverse functions are essential in real-world scenarios where you need to reverse a process. For example, consider a function that converts temperature from Celsius to Fahrenheit: \(F(C) = \frac{9}{5}C + 32\). To convert back from Fahrenheit to Celsius, you use the inverse function \(C(F) = \frac{5}{9}(F - 32)\). Understanding how to verify inverse functions ensures that when you perform a sequence of operations and then "undo" them, you return to your original starting point, which is crucial in fields like computer science (encryption/decryption), economics, and engineering.