21. The graph of $y=\frac{1}{x}$ is reflected in the line $y=1$. The resulting image is reflected in the line $y=-x$.
What is the equation of the final graph?
A $y=\frac{-1}{(x+2)}$
B $y=\frac{1}{(x-1)}$
C $y=\frac{1}{(x-2)}$
D $y=\frac{-1}{(x-1)}$
E $y=\frac{-1}{(x-2)}$
SOLUTION
A
CoMMENTARY
We give two methods for answering this question.
The first quick method is adequate in the context of the SMC where you can assume that one of the options is correct, and you are not required to justify the answer.
The second method uses algebra to calculate the equation of the final graph, and is an example of the kind of answer that you would need to give when fully explained solutions are required.

MeTHOD 1
When $x=1$, we have $\frac{1}{x}=1$, and therefore the point with coordinates $(1,1)$ lies on the graph of $y=\frac{1}{x}$.
The point $(1,1)$ remains fixed when reflected in the line $y=1$, and then is mapped to the point $(-1,-1)$ after reflection in the line $y=-x$.

It follows that the point $(-1,-1)$ lies on the final graph. It is easy to check that the equation of option A gives $y=-1$ when $x=-1$, but that none of the other equations have this property. We can therefore eliminate options B, C, D and E, and hence conclude that the correct option is A.

NoтE
This argument conclusively eliminates the options $\mathrm{B}, \mathrm{C}, \mathrm{D}$ and $\mathrm{E}$ but does not prove that the equation of option $\mathrm{A}$ is the equation of the final graph. It only shows that this is a possibility.

Method 2
Reflection in the line $y=1$ interchanges the points with coordinates $(x, y)$ and $(x, 2-y)$, as shown in the first diagram on the right.
Therefore the image of the graph of $y=\frac{1}{x}$ after this reflection is the graph of $2-y=\frac{1}{x}$.

Reflection in the line $y=-x$ interchanges the points with coordinates $(x, y)$ and $(-y,-x)$, as shown in the second diagram on the right.
Therefore the image of the graph of $2-y=\frac{1}{x}$ after this reflection is the graph of $2-(-x)=\frac{1}{-y}$. This last equation may be rearranged as $y=\frac{-1}{(x+2)}$.