A parameter can destroy invertibility
8 written marks
The function \(f_a\) is defined by \(\displaystyle f_a(x)=\frac{ax+4}{x+1}\), \(x\ne-1\), where \(a\) is real.
Part a · 2 marks
Write the rule in the form \(\displaystyle a+ \frac{c}{x+ 1}\). State exactly when \(f_a\) has an inverse function.
Part b · 3 marks
Given that \(f_a^{-1}(5)=2\), find \(a\) and the inverse rule, including its excluded input.
Part c · 3 marks
For this value of \(a\), let \(g\) be the restriction of \(f_a\) to \(x\ge0\). Find the domain and range of \(g^{-1}\), and evaluate \(\displaystyle g^{- 1}(\frac{21}{4})\).
Complete-answer checking, worked solutions and indicative written criteria.
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