A corner shop prices a delivery item using \(f(x) = 1.2x + 3\), where \(x\) is the wholesale cost in pounds. A loyalty discount is then applied using \(g(x)...
Question 1 Report
A corner shop prices a delivery item using \(f(x) = 1.2x + 3\), where \(x\) is the wholesale cost in pounds. A loyalty discount is then applied using \(g(x) = x - 4\).
- Find \(f^{-1}(x)\). (2)
- Find an expression for \(gf(x)\) in terms of \(x\). (2)
- Given that \(gf(x) = 32\), find the value of \(x\). (1)
Answer Details
An inverse function \( f^{-1}(x) \) undoes what \( f(x) \) does, found by writing \( y=f(x) \) and rearranging to make \( x \) the subject; a composite function \( gf(x) \) means applying \( f \) first and then \( g \) to the result.
(a) Writing \( y = 1.2x+3 \) and rearranging: subtracting 3 from both sides gives \( y-3=1.2x \), then dividing by 1.2 gives \( x = \dfrac{y-3}{1.2} \). Swapping \(x\) and \(y\) to express the inverse in the usual function notation gives \( f^{-1}(x) = \dfrac{x-3}{1.2} \) [2 marks].
(b) \( gf(x) \) means apply \( f \) first, then apply \( g \) to that result: \( gf(x) = g(f(x)) = f(x) - 4 = (1.2x+3) - 4 = 1.2x - 1 \) [2 marks].
(c) Setting \( gf(x) = 32 \) gives \( 1.2x - 1 = 32 \), so adding 1 to both sides gives \( 1.2x = 33 \), and dividing by 1.2 gives \( x = 27.5 \) [1 mark].
The order in \( gf(x) \) always means "\(f\) then \(g\)", reading right to left, which is why the calculation applies \(f\)'s rule first and then subtracts 4 from that result, not the other way round.
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