Steve completed 9 homework problems in class. The function [tex]$p(m)$[/tex] relates the time (in minutes) Steve 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]$p(m)=\frac{m}{6}+9$[/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]$m(p)=54 p+6$[/tex]
B. [tex]$m(p)=54 p-6$[/tex]
C. [tex]$m(p)=6 p-54$[/tex]
D. [tex]$m(p)=6 p+54$[/tex]
Answer (1)
Start with the given function: p ( m ) = 6 m + 9 .
Replace p ( m ) with p : p = 6 m + 9 .
Solve for m in terms of p : m = 6 ( p − 9 ) .
Simplify to find the inverse function: m ( p ) = 6 p − 54 , so the answer is 6 p − 54 .
Explanation
Understanding the Problem We are given the function p ( m ) = 6 m + 9 , which relates the time Steve spends on homework to the number of problems he completes. We want to find the inverse function m ( p ) , which gives the time spent on homework as a function of the number of problems completed.
Isolating m To find the inverse function, we need to solve for m in terms of p . Start by writing the equation as: p = 6 m + 9
Subtracting 9 Subtract 9 from both sides of the equation: p − 9 = 6 m
Multiplying by 6 Multiply both sides of the equation by 6 to isolate m :
6 ( p − 9 ) = m
Distributing Distribute the 6 on the left side: 6 p − 54 = m
The Inverse Function So the inverse function is: m ( p ) = 6 p − 54
Final Answer Comparing this to the given options, we see that the correct answer is C.
Examples
Understanding inverse functions is crucial in many real-world scenarios. For instance, if you know how many items you can produce per hour, the inverse function tells you how many hours it takes to produce a specific number of items. This concept is widely used in manufacturing, logistics, and project management to optimize time and resources.
