How can one determinate the variationt of f(g(x))? Given the two function: f(x)=x^2-2x, g(x)=sqrt{x+1}

Max Macias 2022-08-09 Answered
How can one determinate the variationt of f(g(x))
Given the two functions:
f ( x ) = x 2 2 x
g ( x ) = x + 1
The question is determinate the variation of f(g(x))
We have D f ( g ( x ) ) = ( 1 , + )
For f it's decreasing for x < 1.
Increasing for x > 1
For g its increasing for x > 1
To determinate the variationf of f(g(x))
In interval ( 1 ; + )
We have g is increasing
X > 1 means that g ( x ) > g ( 1 ) so g ( x ) > 0.
So g ( [ 1 ; + ) ) = [ 0 , + ).
But the problem is that f in that interval is increasing and decreasing Im stuck here.
Can one write a methode to answer any question like this theoricaly.
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Answers (1)

Kobe Ortiz
Answered 2022-08-10 Author has 9 answers
Explanation:
You ought to find intervals on which g ( x ) < 1 and on which g ( x ) > 1 and then compute the variation on each interval separately. We have g ( 1 ) = 1 and g is increasing, so h := f g is decreasing on (-1, 0] and increasing on [ 0 , ). Therefore the variance is ( h ( 0 ) h ( 1 ) ) + ( lim x h ( x ) h ( 1 ) ) = .

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