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 &lt; 1.Increasing for x &gt; 1For g its increasing for x &gt; − 1To determinate the variationf of f(g(x))In interval ( − 1 ; + ∞ )We have g is increasing X &gt; − 1 means that g ( x ) &gt; g ( − 1 ) so g ( x ) &gt; 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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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&#039;s decreasing for   x  &amp;lt;  1.Increasing for   x  &amp;gt;  1For g its increasing for   x  &amp;gt;  −  1To determinate the variationf of f(g(x))In interval   (  −  1  ;  +  ∞  )We have g is increasing  X  &amp;gt;  −  1 means that   g  (  x  )  &amp;gt;  g  (  −  1  ) so   g  (  x  )  &amp;gt;  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.

Kobe Ortiz · Accepted Answer

Explanation:You ought to find intervals on which   g  (  x  )  &amp;lt;  1 and on which   g  (  x  )  &amp;gt;  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  )            )        =  ∞  .

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

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