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Alex completed 7 homework problems in class. The function [tex]p(m)[/tex] relates the time (in minutes) Alex spent on his homework at home to the total number of problems he completed. The input is the number of minutes worked. The output is the number of problems completed.
[tex]\[ p(m)=\frac{m}{4}+7 \][/tex]

Which equation represents the inverse function [tex]m(p)[/tex], which uses problems completed as the input and gives minutes worked as the output?

A. [tex]m(p)=28 p-4[/tex]

B. [tex]m(p)=4 p+28[/tex]

C. [tex]m(p)=28 p+4[/tex]

D. [tex]m(p)=4 p-28[/tex]



Answer :

GinnyAnswer
To find the inverse function \( m(p) \) from the given function \( p(m) = \frac{m}{4} + 7 \), follow these steps:

1. Start with the given function:
[tex]\[ p(m) = \frac{m}{4} + 7 \][/tex]

2. Rewrite the equation to express \( m \) in terms of \( p \). Begin by isolating \( \frac{m}{4} \) on one side of the equation:
[tex]\[ p = \frac{m}{4} + 7 \][/tex]

3. Subtract 7 from both sides of the equation to move the constant term:
[tex]\[ p - 7 = \frac{m}{4} \][/tex]

4. To clear the fraction, multiply both sides by 4:
[tex]\[ 4(p - 7) = m \][/tex]

5. Simplify the expression:
[tex]\[ m = 4(p - 7) = 4p - 28 \][/tex]

Therefore, the inverse function, which represents the minutes worked \( m \) as a function of the problems completed \( p \), is:
[tex]\[ m(p) = 4p - 28 \][/tex]

Correspondingly, the correct answer is:
D. [tex]\( m(p) = 4p - 28 \)[/tex]

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