Select the correct answer.

Keith used the following steps to find the inverse of \( f \), but he thinks he made an error.

[tex]\[
\begin{tabular}{|l|c|l|}
\hline Step 1 & \( f(x) = 7x + 5 \) & given \\
\hline Step 2 & \( y = 7x + 5 \) & change \( f(x) \) to \( y \) \\
\hline Step 3 & \( x = 7y + 5 \) & switch \( x \) and \( y \) \\
\hline Step 4 & \( x - 5 = 7y \) & subtract 5 from each side \\
\hline Step 5 & \( \frac{x - 5}{7} = y \) & divide each side by 7 \\
\hline Step 6 & \( \frac{x - 5}{7} = g(x) \) & change \( y \) to \( g(x) \) \\
\hline Step 7 & \( g(x) = \frac{x - 5}{7} \) & switch sides of the equation \\
\hline
\end{tabular}
\][/tex]

A. In step 4, Keith should have added 5 to each side.
B. In step 5, Keith should have multiplied each side of the equation by 7.
C. In step 6, Keith should have switched \( x \) and \( y \).
D. Keith did not make any errors.



Answer :

Let's analyze each step Keith took to find the inverse of the function \( f(x) = 7x + 5 \) to determine whether Keith made any errors.

1. Step 1: \( f(x) = 7x + 5 \)
- This is the given function.

2. Step 2: \( y = 7x + 5 \)
- This step is correct. Here, Keith replaces \( f(x) \) with \( y \) to simplify the notation.

3. Step 3: \( x = 7y + 5 \)
- This step involves switching \( x \) and \( y \), which is the correct first step towards finding the inverse of the function.

4. Step 4: \( x + 5 = 7y \)
- Here, Keith claims he added 5 to each side, but it appears there might be a typo in the description since the correct operation to isolate \( y \) should involve subtracting 5 from \( x \). If we assume this is a typo and the step should be:
\( x - 5 = 7y \), which is appropriate.

5. Step 5: \( \frac{x + 5}{7} = y \)
- This step is incorrect based on the previous step. Instead, after properly correcting Step 4, it should be:
\( \frac{x - 5}{7} = y \)

6. Step 6: \( \frac{x + 5}{7} = g(x) \)
- Assuming the mistake from Step 5 continues, here Keith changes \( y \) to \( g(x) \), which is generally correct, but he used the wrong expression. It should be:
\( \frac{x - 5}{7} = g(x) \)

7. Step 7: \( g(x) = \frac{x + 5}{7} \)
- Switching sides of the equation while keeping the incorrect expression from Step 5.

Even though the steps contain mostly correct algebraic manipulations and logical progressions, there's a critical error in Step 4. Specifically, Keith incorrectly stated the operation. If the steps had been executed correctly with appropriate algebraic manipulation from Step 4 onwards considering the error in addition, Keith did not correctly isolate \( y \) until Step 2 or Step 3.

Therefore, analyzing these steps:

A is the appropriate correction: Keith should have subtracted 5 from each side in Step 4 (resulting in \( x - 5 = 7y \)).

Hence, the correct choice based on the steps provided is:
A. In step 4, Keith should have subtracted 5 from each side.