Question

talib is trying to find the inverse of the function to the right. his work appears beneath it. is his work correct? explain your answer.
$f(x) = -8x + 4$
$y = -8x + 4$
$y - 4 = -8x$
$x = (y - 4)/-8$
$f^{-1}(x) = (y - 4)/-8$

Explanation:

Step1: Recall inverse function steps

To find the inverse of a function \( y = f(x) \), we swap \( x \) and \( y \) and then solve for \( y \). The general steps are: 1. Replace \( f(x) \) with \( y \). 2. Swap \( x \) and \( y \). 3. Solve for \( y \). 4. Replace \( y \) with \( f^{-1}(x) \).

Step2: Analyze Talib's work

• Talib started with \( y=-8x + 4 \), then got \( y - 4=-8x \), then \( x=\frac{y - 4}{-8} \). But when finding the inverse, after solving for \( x \) in terms of \( y \), we need to swap \( x \) and \( y \) (i.e., replace \( x \) with \( y \) and \( y \) with \( x \)) in the final expression for the inverse function.
• Talib's final step has \( f^{-1}(x)=\frac{y - 4}{-8} \), but \( y \) should be replaced with \( x \) here. The correct step after \( x=\frac{y - 4}{-8} \) is to rewrite it as \( y=\frac{x - 4}{-8} \) (by swapping \( x \) and \( y \)) and then \( f^{-1}(x)=\frac{x - 4}{-8}=\frac{-x + 4}{8} \) (simplifying the fraction).

To find the inverse of a function \( y = f(x) \), we follow these steps:

1. Replace \( f(x) \) with \( y \) (Talib did this: \( y=-8x + 4 \)).
2. Swap \( x \) and \( y \) (Talib skipped this crucial step when defining \( f^{-1}(x) \)).
3. Solve for \( y \): From \( y=-8x + 4 \), we get \( y - 4=-8x \), then \( x=\frac{y - 4}{-8} \). After swapping \( x \) and \( y \), we should have \( y=\frac{x - 4}{-8} \), so \( f^{-1}(x)=\frac{x - 4}{-8} \) (or simplified, \( f^{-1}(x)=\frac{-x + 4}{8} \)).

Talib’s error is in the final step: he left \( y \) in the expression for \( f^{-1}(x) \) instead of replacing \( y \) with \( x \) (after swapping \( x \) and \( y \)).

Answer:

Talib's work is incorrect. When finding the inverse function, after solving for \( x \) in terms of \( y \) (\( x=\frac{y - 4}{-8} \)), we must swap \( x \) and \( y \) (replace \( x \) with \( y \) and \( y \) with \( x \)) to get the inverse function. Talib failed to replace \( y \) with \( x \) in the final expression for \( f^{-1}(x) \); the correct inverse should be \( f^{-1}(x)=\frac{x - 4}{-8} \) (or simplified as \( f^{-1}(x)=\frac{-x + 4}{8} \)).