Asymptotes, range, symmetry, a transformed point and an inequality for a rational function

The curve \(C\) has equation \(y=\mathrm{f}(x)\), where \(\mathrm{f}(x)=\dfrac{x^2+4}{x^2-4}\). Part of \(C\) is shown in the diagram below, and \(A\) is the point where \(C\) meets the \(y\)-axis.

1. Find the equation of each asymptote of \(C\).[2]
2. Find the range of \(\mathrm{f}\).[2]
3. The graph of \(y=\mathrm{f}(x)\) is transformed to the graph of \(y=\mathrm{f}(2x+3)\). Find the coordinates of the image of \(A\).[2]
4. State, with a reason, whether \(\mathrm{f}\) is an odd function, an even function, or neither.[2]
5. Using the diagram, or otherwise, solve the inequality \(\dfrac{x^2+4}{x^2-4}>x-1\).[4]