Sofia made a mistake in step [tex]\( 3 \)[/tex]. She should have properly swapped [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and then solved for [tex]\( y \)[/tex]. Specifically, she should have written [tex]\( x = \frac{3y + 4}{8} \)[/tex] and then multiplied both sides by [tex]\( 8 \)[/tex] to isolate [tex]\( y \)[/tex].
Here's the corrected step-by-step solution:
1. Step 1: [tex]\( f(x) = \frac{3x + 4}{8} \)[/tex] (given)
2. Step 2: Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] to get [tex]\( y = \frac{3x + 4}{8} \)[/tex].
3. Correct Step 3: Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse, giving us [tex]\( x = \frac{3y + 4}{8} \)[/tex].
4. Correct Step 4: Multiply both sides by [tex]\( 8 \)[/tex] to get [tex]\( 8x = 3y + 4 \)[/tex].
5. Subtract [tex]\( 4 \)[/tex] from both sides to isolate the term with [tex]\( y \)[/tex]: [tex]\( 8x - 4 = 3y \)[/tex].
6. Divide both sides by [tex]\( 3 \)[/tex] to solve for [tex]\( y \)[/tex]: [tex]\( y = \frac{8x - 4}{3} \)[/tex].
7. Finally, replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex] to express the inverse function:
[tex]\[ f^{-1}(x) = \frac{8x - 4}{3} \][/tex].
So, Sofia's mistake occurred in step [tex]\( 3 \)[/tex], where she should have written [tex]\( x = \frac{3y + 4}{8} \)[/tex] and then multiplied both sides by [tex]\( 8 \)[/tex] to proceed correctly.
