Sofia used these steps to find the inverse of function [tex]\( f \)[/tex].

\begin{tabular}{|c|c|c|}
\hline
Step Number & Function & Step Description \\
\hline
Step 1 & [tex]\( f(x)=\frac{3x+4}{8} \)[/tex] & Given \\
\hline
Step 2 & [tex]\( y=\frac{3x+4}{8} \)[/tex] & Change [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex] \\
\hline
Step 3 & [tex]\( x=\frac{3y+4}{8} \)[/tex] & Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] \\
\hline
Step 4 & [tex]\( 8x=3y+4 \)[/tex] & Multiply each side by 8 \\
\hline
Step 5 & [tex]\( 8x-4=3y \)[/tex] & Subtract 4 from each side \\
\hline
Step 6 & [tex]\( \frac{8x-4}{3}=y \)[/tex] & Divide each side by 3 \\
\hline
Step 7 & [tex]\( \frac{8x-4}{3}=f^{-1}(x) \)[/tex] & Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex] \\
\hline
\end{tabular}

Use this information to complete the statement.

Sofia made a mistake in step [tex]\(\_\_\_\_\)[/tex]. She should have [tex]\(\_\_\_\_\)[/tex].



Answer :

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.