F(x) = {x}^(2) - 4 for all instances of x \leqslant 0 a) show that f has an inverse function {f}^(- 1) b) find dom( {f}^( - 1) ) \: and \: ran( {f}^( - 1) ) c) find {f}^( - 1) (…

Question

`f(x) = {x}^(2) - 4`

for all instances of

`x \leqslant 0`
a) show that f has an inverse function

`{f}^(- 1)`
b) find

`dom( {f}^( - 1) ) \: and \: ran( {f}^( - 1) )`
c) find

`{f}^( - 1) (x)`

​

Answer 1

Given function
`f(x)=x^2-4` find its inverse by substituting x for f(x) and then solving for f(x).

`x=f(x)^2-4\implies f(x)^(-1)=√(x+4)`

Where
`x+4>=0` for x to be real.

So solve the inequality and you will obtain the domain:

`x+4>=0\implies x>=-4\implies x\in[-4,+\infty)`.

Range is equal to the range of square root function,

`y\in[0, +\infty)`.

Hope this helps.