- Start with the given linear function \(y = 6 - \frac{x}{2}\).
- Swap the variables \(x\) and \(y\) to set up the inverse relationship: \(x = 6 - \frac{y}{2}\).
- Rearrange the equation to isolate the variable \(y\), leading to \(\frac{y}{2} = 6 - x\).
- Solve for \(y\) to obtain the explicit inverse function: \(y = 12 - 2x\).
- State the final inverse function notation as \(\boxed{f^{-1}(x) = 12 - 2x}\).
Explanation
- Analysis of the Function and Goal
To find the inverse function of \(y = 6 - \frac{x}{2}\), we must follow a systematic algebraic process. The objective is to express the relationship such that the roles of the input and output variables are reversed. This allows us to determine what function maps the output back to the original input. We start with the given function:
\[y = 6 - \frac{x}{2}\]
- Swapping Variables
The first step in finding an inverse function is to swap the roles of \(x\) and \(y\). By replacing \(x\) with \(y\) and \(y\) with \(x\) in the original equation, we define the relationship for the inverse function:
\[x = 6 - \frac{y}{2}\]
- Isolating the Term with y
Now we must isolate \(y\) to express it in terms of \(x\). First, we rearrange the equation to isolate the term involving \(y\). Adding \(\frac{y}{2}\) to both sides and subtracting \(x\) from both sides, or equivalently rearranging the terms, gives us:
\[\frac{y}{2} = 6 - x\]
- Solving for y
Next, we solve for \(y\) by multiplying both sides of the equation by 2:
\[y = 2 \times (6 - x)\]
\[y = 12 - 2x\]
- Final Notation and Sequence Conclusion
Finally, we express the result using standard inverse function notation. Since the resulting \(y\) represents the inverse function \(f^{-1}(x)\), we conclude:
\[f^{-1}(x) = 12 - 2x\]
The correct sequence of the provided steps is:
- \(y = 6 - \frac{x}{2}\)
- \(x = 6 - \frac{y}{2}\)
- \(\frac{y}{2} = 6 - x\)
- \(y = 12 - 2x\)
- \(f^{-1}(x) = 12 - 2x\)
Thus, the final expression for the inverse is \(\boxed{f^{-1}(x) = 12 - 2x}\).
Examples
Inverse functions are essential in real-world scenarios where you need to reverse a process. For instance, if a temperature scale converter takes Celsius and gives Fahrenheit as \(F = 1.8C + 32\), the inverse function \(C = \frac{F-32}{1.8}\) allows you to convert any Fahrenheit reading back to Celsius. Mastering this process helps in understanding how various systems, algorithms, and physical transformations can be undone to retrieve original data.