CAN SOMEONE PLEASE HELP ME WITH A MATHS QUESTION ASAP !!!

Question

I HAVE INTERNATIONAL BACCALAUREATE TOMORROW AND I DONT UNDERSTAND THIS !!!'

Here is the question:

 

consider the function g, where     g(x)   =    3x

                                                            ----------

                                                         5 + x(squared)

given that the domain of g is x>a find the least value of a such that g has an inverse function .

                                            -

                         

Answer 1

The maximum of g(x) occurs at x = sqrt(5) ~ 2.236  where it has the value g( 5^.5) = 3 sqrt(5) /10 ~ .67

 

A candidate inverse for g can be obtained by textbook methods as ( 3 ± sqrt( 9 - 20 x²) )/ ( 2 x)

 

This is defined as long as x < .67, but is double valued unless a domain restriction is imposed.

 

Choosing a = sqrt(5) means the that domain of g is  [sqrt(5), ∞)  and that the range of g is (0, .67) 

By the principle that the domain of an inverse is the range of the direct function and that the range of the inverse is the domain of the direct function,   this means that the domain of the inverse is (0, .67] and the range is [sqrt)5), ∞)

 This range is obtained using plus sign in the  candidate inverse above.   And, importantly, is single valued due to the domain restriction on the direct function.

 

The idea is that an inverse function is a function, and thus must be single valued.