Keith is trying to find the inverse of the function [tex]\( f(x) = 7x + 5 \)[/tex]. Let's examine each step he took:
1. Step 1: [tex]\( f(x)=7x+5 \)[/tex]
- This is the given function.
2. Step 2: [tex]\( y=7x+5 \)[/tex]
- Correctly converted [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex].
3. Step 3: [tex]\( x=7y+5 \)[/tex]
- Correctly switched [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
4. Step 4: [tex]\( x+5=7y \)[/tex]
- Here Keith made a mistake. Instead of adding 5 to both sides, he should have subtracted 5 from each side. The correct step should be:
[tex]\[
x - 5 = 7y
\][/tex]
5. Step 5: [tex]\( \frac{x-5}{7}=y \)[/tex]
- After correcting Step 4, divide both sides by 7:
[tex]\[
y = \frac{x-5}{7}
\][/tex]
6. Step 6: [tex]\( \frac{x-5}{7}=g(x) \)[/tex]
- Correctly changed [tex]\( y \)[/tex] to [tex]\( g(x) \)[/tex].
7. Step 7: [tex]\( g(x)=\frac{x-5}{7} \)[/tex]
- Correctly switched sides to express [tex]\( g(x) \)[/tex].
Given this analysis, the error occurs in Step 4. The correct operation is to subtract 5 from each side, not add 5.
Thus, the correct answer is:
- D. In step 4, Keith should have subtracted 5 from each side.
