Figure 1 shows the sketch of a curve with equation $y = f(x),\, x \in \mathbb{R}$ - Edexcel - A-Level Maths Pure - Question 7 - 2018 - Paper 1

To find the solution of $f(2x) = 0$, we need to identify the points where the original curve intersects the x-axis. From the information given, $f(x) = 0$ at $x = 5$. Therefore, setting $2x = 5$ gives:

$x = \frac{5}{2} = 2.5.$

Thus, the solution is $x = 2.5$.

The asymptote for the original function is given as $y = 1$. The transformation $y = f(-x)$ reflects the curve across the y-axis. However, the asymptote will remain unchanged. Therefore, the equation of the asymptote remains:

$y = 1.$

For the line $y = k$ to meet the curve $y = f(x)$ at only one point, $k$ must equal the maximum point of the curve or be less than it. Hence, based on the turning point $y = 7$ at $x = 2$, the set of possible values for $k$ is:

$k \leq 7.$

Additionally, since the line cannot intersect the curve at values greater than the maximum point, the final expression is:

$k < 1 \, \text{ or } \, k = 7.$