Question

How do you find its inverse and check your answer and state the domain and the range of `f(x) = 4 /( x+2)` and f^-1?

Answer 1

Explained below

Explanation:

The domain: `D(f)= RR//{-2}`
`x+2!=0`
`x!=-2`

In order to find inverse function the following statement must be true:
For every `x_1//x_2 in D(f): x_1!=x_2 => f(x_1)!=f(x_2)`

Proof by contradiction:
`4/(x_1+2)=4/(x_2+2)`
`cancel4(x_2+cancel2)=cancel4(x_1+cancel2)`
`x_2=x_1`
So now we know that this function is simple and has an inverse function. Let's find it.

`y=4/(x+2)`

`x=4/(y+2)`

`x(y+2)=4`

`y+2=4/x`

`f^-1 : y=4/x-2`

If we want to find `H(f)` we have to find `D(f^-1)`. It's:
`x!=0`
so `H(f)=RR//{0}`

(english is not my native language which means you may use different way to solve it)