2025 May TZ1 P1 Q8

The function \(f\) is defined by \(f(x) = 5(x + 1)(x + 3)\), where \(x \in \mathbb{R}\).

1. Write \(f(x)\) in the form \(a(x - h)^2 + k\), where \(a, h, k \in \mathbb{Z}\).

   [4]

2. Sketch the graph of \(y = f(x)\), showing the values of any intercepts with the axes and the coordinates of the vertex.

   [4]

3. Solve the inequality \(f(x) \leq 40\).

   [4]

The function \(g\) is defined by \(g(x) = \ln{x}\), where \(x \in \mathbb{R}, x > 0\).

4. [3]

   1. Write down an expression for \((f \circ g)(x)\).

   2. Solve the inequality \((f \circ g)(x) \leq 40\).