Let's go through the steps to find the inverse of the function [tex]\( f \)[/tex] and identify any errors:
1. Step 1: [tex]\( f(x)=\frac{3x+4}{8} \)[/tex]
This is the given function.
2. Step 2: [tex]\( y=\frac{3x+4}{8} \)[/tex]
Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex].
3. Step 3: [tex]\( x=\frac{3y+4}{8} \)[/tex]
Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to begin finding the inverse.
4. Step 4: [tex]\( 8x=3y+4 \)[/tex]
Multiply both sides by 8 to clear the fraction.
5. Step 5: [tex]\( 8x-4=3y \)[/tex]
Subtract 4 from both sides to isolate terms involving [tex]\( y \)[/tex] on one side.
6. Step 6: [tex]\( 24x-12=y \)[/tex]
This step is incorrect. Instead of multiplying both sides by 3, we should isolate [tex]\( y \)[/tex] by dividing both sides by 3.
Correct Step 6: [tex]\( y = \frac{8x-4}{3} \)[/tex]
7. Step 7: [tex]\( f^{-1}(x) = \frac{8x-4}{3} \)[/tex]
Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex] to express the inverse function.
Thus, Sofía made a mistake in step 6. She should have divided both sides by 3.
So the correct statement is:
Sofía made a mistake in step 6. She should have divided both sides by 3.
