- A function \(g(x)\) is the inverse of \(f(x)\) if the composition \(f(g(x)) = x\).
- Substituting \(g(x) = \frac{x}{3}\) into \(f(x) = 3x\) gives \(f(g(x)) = 3(\frac{x}{3})\).
- Simplifying this expression results in \(f(g(x)) = x\), which confirms that \(g(x)\) is the inverse of \(f(x)\).
- The correct choice is \(\boxed{f(g(x)) = 3\left(\frac{x}{3}\right)}\).
Explanation
- Understanding the Inverse Function Definition
The problem asks us to determine which expression correctly verifies that \(g(x) = \frac{1}{3}x\) is the inverse of \(f(x) = 3x\). By definition, a function \(g(x)\) is the inverse of \(f(x)\) if the composition of the two functions satisfies both \(f(g(x)) = x\) and \(g(f(x)) = x\). We need to calculate these compositions for the given functions to find the correct verification expression.
- Evaluating f(g(x))
To verify the inverse, let's first evaluate \(f(g(x))\). We take the expression for \(f(x)\), which is \(f(x) = 3x\), and substitute \(g(x) = \frac{x}{3}\) for \(x\):
\[f(g(x)) = 3 \cdot g(x) = 3 \cdot \left(\frac{x}{3}\right)\]
This expression simplifies as follows:
\[f(g(x)) = 3 \cdot \left(\frac{x}{3}\right) = x\]
This matches the requirement \(f(g(x)) = x\), effectively verifying that \(g(x)\) is indeed the inverse of \(f(x)\).
- Identifying the Correct Expression
Now let's examine the provided choices to identify the correct one:
- \(f(g(x)) = 3\left(\frac{x}{3}\right)\): This correctly represents substituting \(g(x)\) into \(f(x)\) and simplifies to \(x\).
- \(g(f(x)) = \left(\frac{1}{3}x\right)(3x)\): This is incorrectly formed as it represents multiplication rather than composition, and it simplifies to \(x^2\), not \(x\).
- \(f(g(x)) = \frac{1}{3}(3x)\): This expression is logically incorrect as it uses the structure of \(g(f(x))\) while labeled as \(f(g(x))\).
- \(g(g(x)) = \frac{1}{3}\left(\frac{1}{3}x\right)\): This represents the composition of \(g\) with itself, which does not verify the inverse property.
Comparing our calculation from the previous step, the expression \(f(g(x)) = 3\left(\frac{x}{3}\right)\) is the correct choice.
Examples
In computer programming, inverse functions are crucial for data encoding and decoding. For instance, if you have an encryption function that multiplies a numeric key by 3, the inverse function (dividing by 3) acts as the decryption process. Verifying these inverses ensures that the original data can be recovered perfectly after processing. Understanding this relationship helps in creating reliable data management systems where data integrity is paramount.