Curve match

Which curve is which, and how would you plan a route to pass between them?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
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The following five functions have been plotted accurately on the chart below

$$y=\sqrt{x}\quad\quad y=x^2\quad\quad y=x\quad\quad y=(x^2+x)/2\quad\quad y=2\sqrt{x}-x$$

Image
Curve match


Which is the $x$-axis and which is the $y$-axis? Which curves correspond to which functions?

Imagine that you wish to plot a route between $(0, 0)$ and $(1, 1)$. Can you find a function which does this without intersecting any of the existing curves, except at the endpoints?

What other curves can you plot which only intersect the existing curves at the end points?