A community library charges a fine of \(f(x) = 0.5x + 2\) pounds for a book returned \(x\) days overdue. Members receive a loyalty discount modelled by \(g(...
Question 1 Report
A community library charges a fine of \(f(x) = 0.5x + 2\) pounds for a book returned \(x\) days overdue. Members receive a loyalty discount modelled by \(g(x) = 0.9x\), applied to the fine.
- Find \(f(6)\), the fine before any discount for a book 6 days overdue. (1)
- Find an expression for \(gf(x)\), giving your answer in its simplest form. (3)
- A member's discounted fine was \(\pounds 9.90\). By solving \(gf(x) = 9.90\), work out how many days overdue the book was. (2)
Answer Details
A composite function \(gf(x)\) means applying \(f\) first, then applying \(g\) to that result, and setting a composite function equal to a known value turns the problem into an ordinary linear equation to solve.
(a) Substituting \(x=6\) into \(f\):
\[f(6)=0.5(6)+2=3+2=\pounds5\] [1 mark](b) Applying \(g\) to the expression for \(f(x)\), since \(gf(x)\) means \(g(f(x))\):
\[gf(x)=g(0.5x+2)=0.9(0.5x+2)=0.45x+1.8\] [3 marks](c) Setting this expression equal to the discounted fine and solving for \(x\):
\[0.45x+1.8=9.9\] \[0.45x=8.1\] \[x=18 \text{ days}\] [2 marks]The order of composition matters here: \(gf(x)\) applies the overdue fine formula \(f\) first, then the loyalty discount \(g\) to that fine, which correctly models a discount applied on top of an already-calculated fine.
Download The App On Google Playstore
Everything you need to excel in your exams
