Let’s carefully review Talib’s steps and see if his calculation of the inverse function is correct.
Given the function [tex]\( f(x) = -8x + 4 \)[/tex], to find its inverse, we follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = -8x + 4
\][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[
x = -8y + 4
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
- First, isolate the term involving [tex]\( y \)[/tex] by subtracting 4 from both sides of the equation:
[tex]\[
x - 4 = -8y
\][/tex]
- Then, divide both sides by -8 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x - 4}{-8}
\][/tex]
4. Express the inverse function:
[tex]\[
f^{-1}(x) = \frac{x - 4}{-8}
\][/tex]
Talib's work ends with [tex]\( f^{-1}(x) = \frac{(y-4)}{-8} \)[/tex], which appears to be a mistake in notation. The correct inverse function should be stated in terms of [tex]\( x \)[/tex] instead of [tex]\( y \)[/tex]. The correct notation would be:
[tex]\[
f^{-1}(x) = \frac{x - 4}{-8}
\][/tex]
Therefore, Talib’s steps are almost correct but his final notation should reflect [tex]\( f^{-1}(x) \)[/tex] instead of [tex]\( \frac{(y-4)}{-8} \)[/tex]. Tetchnically, he switched [tex]\( x \)[/tex] and [tex]\( y \)[/tex] correctly before solving for [tex]\( y \)[/tex]. Thus, if the final step is explicitly written in terms of [tex]\( x \)[/tex], his work would be correct.
Given the choices:
- No, Talib's work is not correct. He should have correctly. He should have added 4 to both sides. switched [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and then
- Yes, Talib's work is correct.
The correct response should be:
Yes, Talib's work is correct. (Assuming an understanding that the final equation should properly be expressed in terms of [tex]\( x \)[/tex])
