Sketched below are the graphs of the functions defined by: $f(x) = -\frac{k}{x - 4}(x + 2)$ and $g(x) = -\frac{k}{x} + q$ - NSC Technical Mathematics - Question 4 - 2023 - Paper 1

To find the coordinates of point D, we analyze the turning point of the graph of $f(x)$. The coordinates of D are:

• D: $(1, 9)$.

The range of the function $f$ is determined based on its behavior. Since $f(x)$ approaches positive infinity as it approaches the x-intercepts and negative infinity otherwise, we have:

• Range of f: $y \in (-\infty, 9]$.

The asymptotes of the function $g(x)$ can be expressed as:

• Vertical asymptote: $x = 0$.
• Horizontal asymptote: $y = 9$.

From the provided graph and function definitions, setting $g(0) \rightarrow 9$, we can derive:

• Value of k: $k = 18$.

To find where $g(x) \leq 0$, we analyze the function and derive:

• The function is less than or equal to zero for $-2 \leq x \leq 0$.