Figure 2 shows a sketch of the curve with the equation $y = f(x)$, $x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 7 - 2010 - Paper 5

To find the transformed coordinates of point $A(3, -4)$:

1. First apply the horizontal transformation,
   • The x-coordinate will change as follows: $x = 3$ becomes $\frac{1}{2}(3) = 1.5$.
2. Then, apply the vertical transformation:
   • The y-coordinate will be $y = 2(-4) = -8$.

Thus, the transformed point is $A'(1.5, -8)$.

The curve $y = f(|x|)$ takes the left part of the curve and reflects it to the right side. The turning point $A(3, -4)$ remains at $(-3, -4)$ on the left since $|x|$ maps negative x to positive. Both the points where the curve cuts the y-axis at $(0, 5)$ and turning points at $(3, -4)$ and $(-3, -4)$ should be explicitly marked on the sketch.

Since the curve $y = f(x)$ is a translation of $y = x^2$, we need to determine the vertex shift:

If $f(x)$ is translated down to have a turning point of $A(3, -4)$, we can express it as:

$f(x) = (x - 3)^2 - 4$

This equation represents a parabola with vertex at $(3, -4)$. The function opens upwards.

The function $f(x) = (x - 3)^2 - 4$ is a quadratic function that opens upwards. Since it has a turning point and is not one-to-one (passes the vertical line test), it does not have an inverse function. For a function to have an inverse, it must be bijective (both one-to-one and onto).