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Inverting Rational Functions

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

These questions concerning rational functions were inspired by responses to the function machine project Steve's Mapping. In this problem use the definition that a rational function is defined to be any function which can be written as the ratio of two polynomial functions.

Consider these two rational functions

$$
f(x)=\frac{2x+9}{x+2}\quad\quad g(x)=\frac{9-2x}{x-2}
$$
Show that they are inverses of each other, in that
$$
g(f(x))=f(g(x))=x
$$
What happens for the values $x=\pm 2$?

Can you invert the rational function

$$
h(x)=\frac{x-7}{2x+1}
$$

Do rational functions always have inverse functions? Why?

In the examples given here, the inverses of our rational functions were also rational functions. Will this be the case more generally? Why not explore more generally or try to find inverse pairs of rational functions?

As you consider these rational functions, many questions might emerge in your mind such as: "do rational functions have fixed points?" or "Is there a relationship between the asymptotes in a function and the zeroes of its inverse?". Why not make a note of these questions and ask your teacher, yourself or your friends to try to solve them?