A phone tariff gives the monthly cost, in pounds, as \(f(x) = 12 + 0.05x\), where \(x\) is the number of minutes used that month, under the provider's stand...
Question 1 Report
A phone tariff gives the monthly cost, in pounds, as \(f(x) = 12 + 0.05x\), where \(x\) is the number of minutes used that month, under the provider's standard price plan.
- Find \(f^{-1}(x)\), the inverse function of \(f\). (2)
- A customer's bill one month was \(\pounds 19.70\). Use \(f^{-1}(x)\) to find the number of minutes they used. (1)
Answer Details
Finding the inverse function means writing the rule the other way round, from cost back to minutes used; once \(f^{-1}\) is found algebraically, it can be applied directly to a specific bill amount to recover the minutes used, without solving the original equation from scratch.
- Writing \(y=f(x)=12+0.05x\) and rearranging to make \(x\) the subject: \[ y-12 = 0.05x \] \[ x = \frac{y-12}{0.05} = 20y-240 \] so \[ f^{-1}(x) = 20x-240 \] [2 marks]
- \[ f^{-1}(19.70) = 20\times19.70-240 = 394-240 = 154 \text{ minutes} \] [1 mark]
Applying \(f^{-1}\) to a bill amount directly gives the minutes used, which is exactly what makes an inverse function useful: it undoes the original function, turning an output (cost) back into the input (minutes) that produced it.
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