The functions $f$ and $g$ are given by $f(x) = x + \frac{1}{x}$ for $x > 0$, and $g(x) = ax + 1$ for $x \in \mathbb{R}$, with $a$ a constant.
(a)[1]
Obtain an expression for $gf(x)$.
(b)[2]
Using $gf(2) = 11$, determine the value of $a$.
(c)[1]
Since the graph of $y = f(x)$ has a minimum point at $x = 1$, explain whether $f$ has an inverse.
(d)[3]
Now assume $a = 5$. Find and simplify an expression for $g^{-1}f(x)$.
(e)[1]
Explain why the composite function $fg$ cannot be formed.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Correct expression $a\left(x+\frac{1}{x}\right)+1$” …