2025 NOV TZ1 P1 Q7

Consider the function \(f(x) = 2x + 3\) where \(x \in \mathbb{R}\).

1. Find

   [3]

   1. \(f(2)\);

   2. \(f^{-1}(x)\).

The following diagram shows part of the graph of a linear function \(h\).
The graph has \(x\)-intercept \(2\) and \(y\)-intercept \(-1\).

2. Write down an expression for \(h(x)\).

   [2]

3. Solve the equation \(h^{-1}(x) = -2\).

   [2]

Consider the function \(g(x) = mx + c\), where \(x \in \mathbb{R}\) and \(m, c \in \mathbb{Q}\)

4. Given that \(h(x) = (f^{-1} \circ g)(x)\), find the value of \(m\) and the value of \(c\).

   [4]

A function \(k\) exists such that \(h(k(x)) = x\).

5. Find an expression for \(k(x)\).

   [2]

6. State the single transformation that maps the graph of \(y = k(x)\) onto the graph of \(y = h(x)\).

   [1]