The function f is defined by $$f: x \mapsto \frac{3 - 2x}{x - 5}, \; x \in \mathbb{R}, \; x \neq 5$$ (a) Find $f^{-1}(x)$ - Edexcel - A-Level Maths Pure - Question 6 - 2011 - Paper 4

The range of the function $g$ can be determined from its graph. From $(−1,−9)$ to $(2,0)$, the values range from $-9$ to $0$. From $(2,0)$ to $(8,4)$, the values continue from $0$ up to $4$. Thus, the complete range is: $[-9, 4]$

To find $g(2)$, we observe from the linear function segment connecting $(2,0)$ that: $g(2) = 0$

To find $fg(8)$, we first determine $g(8)$ from the graph. From the graph, we see: $g(8) = 4$
Then, we compute $f(g(8))= f(4)$: Substituting into the function: $f(4) = \frac{3 - 2(4)}{4 - 5} = \frac{3 - 8}{-1} = \frac{-5}{-1} = 5$
Thus, $fg(8) = 5$.

For the graph of $y = |g(x)|$,

• The section from $(-1, -9)$ to $(2, 0)$ will reflect above the x-axis, resulting from calculating the absolute value.
• From $(2, 0)$ to $(8, 4)$, the section remains unchanged since it is already positive.

In summary, sketching should show the transposed portion from $(-1, -9)$ to $(2, 0)$ as $(-1, 9)$ and $(2, 0)$, while $(2, 0)$ to $(8, 4)$ remains unchanged.

The graph of the inverse function $y = g^{-1}(x)$ can be sketched by swapping the $(x,y)$ coordinates of the original function's graph. Therefore, the segments will reflect:

• The segment from $(−9,−1)$ becomes $(−1,−9)$
• The segment from $(0,2)$ becomes $(2,0)$, and
• The segment from $(4,8)$ becomes $(8,4)$.

Sketching should illustrate these coordinates clearly on the axes.

The domain of the inverse function $g^{-1}$ corresponds to the range of the original function $g$. Since we found the range to be: $[-9, 4]$
The domain of $g^{-1}$ is therefore: $-9 \leq x \leq 4$