Verified Answer :
- Set the function equal to \(y\): \(y = x + 3\).
- Solve for \(x\) in terms of \(y\): \(x = y - 3\).
- Swap the variables to define the inverse function: \(f^{-1}(x) = x - 3\).
- The final answer is \(\boxed{f^{-1}(x) = x - 3}\).
Explanation
- Understanding the Concept of an Inverse Function
The objective is to find the inverse of the linear function \(f(x) = x + 3\). An inverse function \(f^{-1}(x)\) is a function that reverses the operation performed by \(f(x)\). Specifically, if \(f(x) = y\), then \(f^{-1}(y) = x\). For a function to have an inverse, it must be a bijection, meaning it is both one-to-one and onto. Since \(f(x) = x + 3\) is a linear function with a non-zero slope (specifically a slope of 1), it maps every input to a unique output and covers all real numbers, so an inverse exists.
- Solving for x in terms of y
To find the inverse, we start by setting the function equal to \(y\):
\[y = x + 3\]
Now, our goal is to isolate \(x\) to express it in terms of \(y\). Subtracting \(3\) from both sides of the equation yields:
\[y - 3 = x\]
This shows that if we have a value \(y\), the original input \(x\) is obtained by subtracting \(3\).
- Defining the Inverse Function
By the definition of an inverse function, we now swap the variables \(x\) and \(y\) to express the inverse function \(f^{-1}\) in the standard form with \(x\) as the independent variable. Replacing \(y\) with \(x\) in our result \(x = y - 3\), we obtain:
\[f^{-1}(x) = x - 3\]
- Verifying the Inverse Function
To verify that \(f^{-1}(x) = x - 3\) is indeed the inverse, we check if \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
First, \(f(f^{-1}(x)) = f(x - 3) = (x - 3) + 3 = x\).
Second, \(f^{-1}(f(x)) = f^{-1}(x + 3) = (x + 3) - 3 = x\).
Both conditions are satisfied, confirming the inverse function is correct. Therefore, the inverse of \(f(x) = x + 3\) is \(f^{-1}(x) = x - 3\).
Examples
Inverse functions are essential in situations where you need to undo an operation. For instance, consider a scenario where a shopkeeper adds a fixed cost (a service fee) to every item, represented by the function \(f(x) = x + 3\), where \(x\) is the original price and \(f(x)\) is the final price. If you only have the final price and need to determine the original price of an item, you would use the inverse function \(f^{-1}(x) = x - 3\). This helps in accounting, price analysis, and reversing algorithmic processes.