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

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Respuesta :

r3t40

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

[tex]x=f(x)^2-4\implies f(x)^{-1}=\sqrt{x+4}[/tex]

Where [tex]x+4>=0[/tex] for x to be real.

So solve the inequality and you will obtain the domain:

[tex]x+4>=0\implies x>=-4\implies x\in[-4,+\infty)[/tex].

Range is equal to the range of square root function,

[tex]y\in[0, +\infty)[/tex].

Hope this helps.

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