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.
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.
