one to one functions

Question

The one-to-one function g is defined below.

g(x)=5x/3x-7

Find g^-1(x), where g^-1 is the inverse of g.
Also state the domain and range of g^-1 in interval notation.

g^-1(x) = ?
Domain of g^-1 = ?
Range of g^-1 = ?

Answer 1

You have been given g(x) = 5x/(3x-7) ...NOTE that I have supplied the missing grouping symbols for the multi-term denominator.

 

Exchanging veriables: x = 5y/(3y-7); cross-multiplying gives 3xy - 7x = 5y.

 

factoring gives y(3x-5) = 7x.

 

Solving for y & replacing that by g^-1(x) gives g^-1(x)= 7x/(3x-5).

 

Domain of g: all Reals except 7/3; RANGE of g^-1 is the same.

 

Domain og g^-1: all Reals except 5/3; RANGE of g is the same.