- Substitute \(g(x) = \frac{x}{3} + 3\) into \(f(x) = 3x - 9\) to find \(f(g(x)) = 3(\frac{x}{3} + 3) - 9 = x\).
- Substitute \(f(x) = 3x - 9\) into \(g(x) = \frac{x}{3} + 3\) to find \(g(f(x)) = \frac{3x - 9}{3} + 3 = x\).
- Since both compositions equal \(x\), the functions \(f\) and \(g\) are inverses.
- Final answers: (a) \(f(g(x)) = \boxed{x}\), (b) \(g(f(x)) = \boxed{x}\), (c) Inverse: \(\boxed{\text{Yes}}\).
Explanation
- Analysis of the Functions
We are given two functions: \(f(x) = 3x - 9\) and \(g(x) = \frac{x}{3} + 3\). Our objective is to determine their composite functions \(f(g(x))\) and \(g(f(x))\), and to check whether these functions are inverses of each other. Two functions are inverses if and only if their composition results in the identity function, \(I(x) = x\), in both directions.
- Computing \(f(g(x))\)
To find \(f(g(x))\), we substitute the expression for \(g(x)\) into \(f(x)\). Specifically, we take the formula for \(f(x)\) and replace every occurrence of \(x\) with \(\frac{x}{3} + 3\):
\[f(g(x)) = 3\left(\frac{x}{3} + 3\right) - 9\]
Distributing the 3 inside the parentheses:
\[f(g(x)) = 3 \cdot \frac{x}{3} + 3 \cdot 3 - 9 = x + 9 - 9\]
Simplifying the expression, we get:
\[f(g(x)) = x\]
- Computing \(g(f(x))\)
To find \(g(f(x))\), we substitute the expression for \(f(x)\) into \(g(x)\). Specifically, we take the formula for \(g(x)\) and replace every occurrence of \(x\) with \(3x - 9\):
\[g(f(x)) = \frac{3x - 9}{3} + 3\]
We can divide each term in the numerator by 3:
\[g(f(x)) = \frac{3x}{3} - \frac{9}{3} + 3 = x - 3 + 3\]
Simplifying the expression, we get:
\[g(f(x)) = x\]
- Determining if the Functions are Inverses
Based on our findings, \(f(g(x)) = x\) and \(g(f(x)) = x\). Since both compositions of the functions \(f\) and \(g\) result in the identity function \(x\), we conclude that \(f\) and \(g\) are indeed inverses of each other.
The results are:
A\(f(g(x)) = \boxed{x}\)
B\(g(f(x)) = \boxed{x}\)
CYes, they are inverses. \(\boxed{\text{Yes}}\)
Examples
In many real-world scenarios, we deal with inverse processes. For example, consider a currency converter where \(f(x)\) converts dollars into euros, and \(g(x)\) converts euros back into dollars. If both operations combined result in the original amount of dollars (\(g(f(x)) = x\)) and the original amount of euros (\(f(g(x)) = x\)), then these functions are perfect inverses of each other. Understanding inverses is crucial for reversing operations, decoding messages, or solving equations efficiently.