The diagram shows the graph of the function $f(x) = 5x - x^2$ in the domain $0 \leq x \leq 5$, $x \in \mathbb{R}$ - Leaving Cert Mathematics - Question 5 - 2015

To find the points of intersection of the functions $f(x)$ and $g(x)$, we set them equal to each other:

$5x - x^2 = x + 3$

Rearranging gives:

$-x^2 + 4x - 3 = 0$

Factoring or using the quadratic formula, we can solve for $x$:

1. The quadratic formula is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = -1, b = 4, c = -3$.

2. Calculating the discriminant: $D = 4^2 - 4(-1)(-3) = 16 - 12 = 4$

3. Thus: $x = \frac{-4 \pm 2}{-2}$ This gives $x = 3$ and $x = 1$.

4. Substitute $x$ back to find $y$: For $x = 1$:
   $g(1) = 1 + 3 = 4$ For $x = 3$:
   $g(3) = 3 + 3 = 6$ Hence, the co-ordinates are $A(1, 4)$ and $B(3, 6)$.