- Set the function \(f(x) = 3x + 2\) as \(y = 3x + 2\).
- Subtract \(2\) from both sides to get \(y - 2 = 3x\).
- Divide by \(3\) to isolate \(x\), resulting in \(x = \frac{y - 2}{3}\).
- Replace \(y\) with \(x\) to obtain the inverse function \(\boxed{f^{-1}(x) = \frac{x-2}{3}}\).
Explanation
- Analyzing the Function and Inverse Definition
To find the inverse of the function \(f(x) = 3x + 2\), we look for a function \(f^{-1}(x)\) such that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). Given the function \(f(x) = 3x + 2\), we can treat the expression as a linear equation representing the relationship between an input \(x\) and an output \(y\). Specifically, we have \(y = 3x + 2\). The inverse function essentially reverses this process, taking the output \(y\) and mapping it back to the original input \(x\).
- Solving for the Inverse Function
To determine the expression for \(f^{-1}(x)\), we begin with the equation \(y = 3x + 2\). Our goal is to isolate \(x\) to express it in terms of \(y\). First, we subtract 2 from both sides of the equation:
\[y - 2 = 3x\]
Next, we divide both sides by 3 to solve for \(x\):
\[x = \frac{y - 2}{3}\]
This shows that if the output is \(y\), the original input \(x\) must have been \(\frac{y - 2}{3}\). To express this as a function of \(x\), we replace \(y\) with \(x\):
\[f^{-1}(x) = \frac{x - 2}{3}\]
- Identifying the Correct Inverse Function
Comparing our derived expression \(f^{-1}(x) = \frac{x - 2}{3}\) with the provided options:
- \(f^{-1}(x) = \frac{x-3}{2}\)
- \(f^{-1}(x) = \frac{x+3}{2}\)
- \(f^{-1}(x) = \frac{x-2}{3}\)
- \(f^{-1}(x) = \frac{x+2}{3}\)
The derived inverse function corresponds to the third option. Thus, the inverse function is \(\boxed{f^{-1}(x) = \frac{x-2}{3}}\).
Examples
Inverse functions are commonly used in everyday life to reverse operations. For example, if a currency converter uses the function \(f(x) = 1.1x\) to convert Euro to USD, the inverse function \(f^{-1}(x) = \frac{x}{1.1}\) can be used to convert USD back to Euro. Similarly, in temperature conversion, if you have a function to convert Celsius to Fahrenheit, the inverse allows you to convert Fahrenheit back to Celsius. Understanding how to find inverses helps in solving for unknown variables when the output of a process is already known.