## Things you might have noticed

None of the following sketches is an accurate graph of the function $f(x)=\dfrac{x-5}{x^2-2x-3}.$

But which of these could be a sketch graph of $f(x)$? Try to explain why some of the sketches can’t represent this function.

We will discuss how thinking about various questions can help us to decide which graphs could or could not be $y=f(x).$ You might have considered some of these questions, but perhaps in a different order, or you may not have needed to consider all of these points. There are connections between some of the questions, but we will not assume that graphs have been eliminated at any previous stage, so the sections can be read in any order. You might find it helpful to have the graphs in front of you as you read.

To get started it may be helpful to think about the graphs of the functions in the numerator and denominator of $f(x)$ and consider how the behaviour of these functions determines the behaviour of $f(x).$

• Which sketch graph do you think is the best representation of this function?

Once you have decided which graph(s) could represent this function, try to label key features, such as points where the graph crosses the axes. You could also try to sketch a graph that you think is a bit more true to the function than these graphs.