The function f is defined by $$f : x \mapsto \frac{3 - 2x}{x - 5}, \quad x \in \mathbb{R}, \quad x \neq 5$$ (a) Find $f^{-1}(x)$ - Edexcel - A-Level Maths Pure - Question 7 - 2011 - Paper 4

The function g is defined from $(-1, -9)$ to $(2, 0)$ and from $(2, 0)$ to $(8, 4)$. Therefore, the range of $g$ can be written as:

$\text{Range of } g = \{-9 \leq y \leq 4\}.$

From the graph provided, we observe that at $x = 2$, the value of $g(2)$ is:

$g(2) = 0.$

First, we need to find $g(8)$ from the linear section of the graph, which gives us:

$g(8) = 4.$

Now, substituting this into $f$, we find:

$f(g(8)) = f(4) = \frac{3 - 2(4)}{4 - 5} = \frac{3 - 8}{-1} = 5.$

Thus, $fg(8) = 5.$

1. Sketch of $y = |g(x)|$:

   • Transform the negative values of $g(x)$ (i.e., $y$ values) into positive values, resulting in a V-shaped graph that touches the x-axis at coordinates $(2, 0)$ and $(8, 4)$.
   • Points of intersection with the axes will be at $(0, 9)$ and $(2, 0)$.

2. Sketch of $y = g^{-1}(x)$:

   • To sketch this function, reflect the graph of $g(x)$ about the line $y = x$.
   • The coordinates on the sketch where it meets the axes can be marked as $(0, 2)$ and $(4, 8)$.

The domain of $g^{-1}$ corresponds to the range of $g$. Therefore, the domain of the inverse function is:

$\text{Domain of } g^{-1} = \{-9 \leq x \leq 4\}.$