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## 'Inverting Rational Functions' printed from http://nrich.maths.org/

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?