How do you use composition of functions to show that f(x)=(2+x)/x and f^-1(x) = 2/(x-1) are inverses?

1 Answer
Karthik G
Jan 13, 2016

Step by step explanation is given below.

Explanation:

To prove f(x) and f^-1(x) we need to show
(fcirc f^-1)(x)=x and (f^-1circf)(x)=x

f(x)=(2+x)/x and f^-1(x) = 2/(x-1)

(fcircf^-1)(x)

= (2+2/(x-1))/(2/(x-1))

=((2(x-1))/(x-1)+2/(x-1))/(2/(x-1))

=(2(x-1)+2)/(x-1) xx (x-1)/2

=(2x-2+2)/(x-1)xx(x-1)/2

=(2x)/(x-1)xx(x-1)/2

=(cancel(2)x)/cancel(x-1)xxcancel(x-1)/cancel(2)

=x

We got that (fcirc f^-1)(x)=x

Similarly, we can prove (f^-1circf)(x)=x
(f^-1circf)(x)

=2/((2+x)/x - 1)

=2/((2+x)/x- x/x)

=2/((2+x-x)/x)

=2/((2+cancel(x)-cancel(x))/x)

=2/(2/x)

=2/1 xx x/2

=cancel(2)/1xxx/cancel(2)

=x

Therefore, (f^-1circf)(x)=x

Thus were are able to show that the functions are inverses of each other.

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