- Identify the function: \(f(x) = -8x + 4\).
- Solve for \(x\) in terms of \(y\): \(x = \frac{y-4}{-8}\).
- Interchange variables to define the inverse function: \(f^{-1}(x) = \frac{x-4}{-8}\).
- Verify the result by composing the function with its inverse, confirming the work is correct: \(\boxed{f^{-1}(x) = \frac{x-4}{-8}}\).
Explanation
- Deriving the Inverse Function
The function provided is \(f(x) = -8x + 4\). To find the inverse function \(f^{-1}(x)\), we follow the standard algebraic process of expressing \(x\) in terms of \(y\) where \(y = f(x)\) and then interchanging the variables. First, we set the function equal to \(y\): \[y = -8x + 4.\] Next, we isolate the \(x\) term by subtracting 4 from both sides: \[y - 4 = -8x.\] Then, we divide by \(-8\) to solve for \(x\): \[x = \frac{y - 4}{-8}.\] To express this as a function of \(x\), we interchange the variables \(x\) and \(y\), which gives \(y = \frac{x - 4}{-8}\). Thus, the inverse function is \(f^{-1}(x) = \frac{x - 4}{-8}\).
- Verifying the Inverse Function
We verify the result by checking if the composition of the function and its inverse yields the identity function, i.e., \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
First, we compute \(f(f^{-1}(x))\):
\[f\left(\frac{x - 4}{-8}\right) = -8\left(\frac{x - 4}{-8}\right) + 4 = (x - 4) + 4 = x.\]
Next, we compute \(f^{-1}(f(x))\):
\[f^{-1}(-8x + 4) = \frac{(-8x + 4) - 4}{-8} = \frac{-8x}{-8} = x.\]
Both conditions are satisfied, confirming that the derived expression for the inverse function is correct. Comparing this with Talib's work, we see that all his steps are mathematically sound and reach the correct conclusion.
Examples
Inverse functions are widely used in fields like computer programming and cryptography. For example, if a function encrypts a message by applying a mathematical transformation (like a series of operations), the decryption process is simply the inverse of that function. If you know the original transformation rule, you can find the inverse to "undo" the process and recover the original data. This ensures that information can be securely encoded and then restored by an authorized user who knows the inverse key.