if-f-x-x-2-1-and-g-x-1-x-2-3-find-domain-function-g-f-x-

Question Number 83662 by jagoll last updated on 05/Mar/20

if f (x) = \sqrt{x^{2} - 1} and g (x) = \frac{1}{\sqrt{x^{2} - 3}}

find domain function

(g ∙ f) (x)

Commented by MJS last updated on 05/Mar/20

f (g (x)) or g (f (x)) ?

both interpretations of (g \circ f) are common

Commented by jagoll last updated on 05/Mar/20

yes sir g (f (x))

Commented by mathmax by abdo last updated on 05/Mar/20

g o f (x) = g (f (x)) = \frac{1}{\sqrt{{(f (x))}^{2} - 3}} = \frac{1}{\sqrt{x^{2} - 1 - 3}} = \frac{1}{\sqrt{x^{2} - 4}}

x^{2} - 4 > 0 \Rightarrow∣ x ∣> 2 \Rightarrow x > 2 o r x < - 2 \Rightarrow D_{g o f} =] - \infty, - 2 [\cup] 2, + \infty [

Answered by MJS last updated on 05/Mar/20

f (g (x)) = \sqrt{\frac{4 - x^{2}}{x^{2} - 3}}

3 < x^{2} ⩽ 4 \Rightarrow - 2 ⩽ x < - \sqrt{3} \lor \sqrt{3} < x ⩽ 2

g (f (x)) = \frac{1}{\sqrt{x^{2} - 4}}

x^{2} - 4 > 0 \Rightarrow x^{2} > 4 \Rightarrow x < - 2 \lor x > 2

Commented by jagoll last updated on 05/Mar/20

yess

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