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Sagot :
Let's go through Keith's steps carefully to check his work on finding the inverse function of [tex]\( f(x) = 7x + 5 \)[/tex].
Step-by-step solution:
1. Step 1: [tex]\( f(x) = 7x + 5 \)[/tex]
This is the function given to us.
2. Step 2: [tex]\( y = 7x + 5 \)[/tex]
Replacing [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] is a standard step in finding the inverse function.
3. Step 3: [tex]\( x = 7y + 5 \)[/tex]
Switching [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is also correct as we seek the inverse function [tex]\( y \)[/tex] (now called [tex]\( g(x) \)[/tex]), which essentially reverses the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the original function.
4. Step 4: [tex]\( x + 5 = 7y \)[/tex]
Here Keith has made an algebraic manipulation; however, the correct step to isolate [tex]\( y \)[/tex] would be to subtract 5 from both sides, not add. This step should have been:
[tex]\[ x - 5 = 7y \][/tex]
5. Step 5: [tex]\(\frac{x + 5}{7} = y\)[/tex]
Since the correct step 4 should have been [tex]\( x-5 = 7y \)[/tex], dividing by 7 would give:
[tex]\[ y = \frac{x - 5}{7} \][/tex]
6. Step 6: [tex]\(\frac{x + 5}{7} = g(x) \)[/tex]
Substituting [tex]\( y \)[/tex] with [tex]\( g(x) \)[/tex] is typical when expressing the inverse function. However, since the expression in Step 5 was incorrect, this substitution, while typical, is based on an incorrect prior step.
7. Step 7: [tex]\( g(x) = \frac{x + 5}{7} \)[/tex]
Switching sides of the equation is usually done for clarity, displaying the function [tex]\( g(x) \)[/tex].
Given the steps and the analysis, we see that Keith's initial mistake happened at Step 4 where he added instead of subtracting 5. Thus, Keith’s correct working should look like this:
1. [tex]\( f(x) = 7x + 5 \)[/tex]
2. [tex]\( y = 7x + 5 \)[/tex]
3. [tex]\( x = 7y + 5 \)[/tex]
4. [tex]\( x - 5 = 7y \)[/tex] (subtract 5 from both sides)
5. [tex]\(\frac{x - 5}{7} = y \)[/tex] (divide both sides by 7)
6. [tex]\( g(x) = \frac{x - 5}{7} \)[/tex] (replace [tex]\( y \)[/tex] with [tex]\( g(x) \)[/tex])
Keith should have subtracted 5 in step 4, not added.
Therefore, the correct answer: B. In step 4, Keith should have subtracted 5 from each side.
Step-by-step solution:
1. Step 1: [tex]\( f(x) = 7x + 5 \)[/tex]
This is the function given to us.
2. Step 2: [tex]\( y = 7x + 5 \)[/tex]
Replacing [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] is a standard step in finding the inverse function.
3. Step 3: [tex]\( x = 7y + 5 \)[/tex]
Switching [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is also correct as we seek the inverse function [tex]\( y \)[/tex] (now called [tex]\( g(x) \)[/tex]), which essentially reverses the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the original function.
4. Step 4: [tex]\( x + 5 = 7y \)[/tex]
Here Keith has made an algebraic manipulation; however, the correct step to isolate [tex]\( y \)[/tex] would be to subtract 5 from both sides, not add. This step should have been:
[tex]\[ x - 5 = 7y \][/tex]
5. Step 5: [tex]\(\frac{x + 5}{7} = y\)[/tex]
Since the correct step 4 should have been [tex]\( x-5 = 7y \)[/tex], dividing by 7 would give:
[tex]\[ y = \frac{x - 5}{7} \][/tex]
6. Step 6: [tex]\(\frac{x + 5}{7} = g(x) \)[/tex]
Substituting [tex]\( y \)[/tex] with [tex]\( g(x) \)[/tex] is typical when expressing the inverse function. However, since the expression in Step 5 was incorrect, this substitution, while typical, is based on an incorrect prior step.
7. Step 7: [tex]\( g(x) = \frac{x + 5}{7} \)[/tex]
Switching sides of the equation is usually done for clarity, displaying the function [tex]\( g(x) \)[/tex].
Given the steps and the analysis, we see that Keith's initial mistake happened at Step 4 where he added instead of subtracting 5. Thus, Keith’s correct working should look like this:
1. [tex]\( f(x) = 7x + 5 \)[/tex]
2. [tex]\( y = 7x + 5 \)[/tex]
3. [tex]\( x = 7y + 5 \)[/tex]
4. [tex]\( x - 5 = 7y \)[/tex] (subtract 5 from both sides)
5. [tex]\(\frac{x - 5}{7} = y \)[/tex] (divide both sides by 7)
6. [tex]\( g(x) = \frac{x - 5}{7} \)[/tex] (replace [tex]\( y \)[/tex] with [tex]\( g(x) \)[/tex])
Keith should have subtracted 5 in step 4, not added.
Therefore, the correct answer: B. In step 4, Keith should have subtracted 5 from each side.
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