Composing a function with its antiderivative What does it mean that, given a function and its antid

junoonib89p4

junoonib89p4

Answered question

2022-05-10

Composing a function with its antiderivative
What does it mean that, given a function and its antiderivative, if I make the composition g ( h ( x ) ) = h ( g ( x ) ) = α ( x )? I mean, I was thinking that α ( x ) could be some sort of identity function.
For example: Given, g ( x ) = x 3 3 , h ( x ) = x 2 , then g ( h ( x ) ) = x 6 3 = h ( g ( x ) ) where x 6 3 = α ( x )

Answer & Explanation

Lucille Melton

Lucille Melton

Beginner2022-05-11Added 18 answers

Step 1
Since h ( x ) = x 2 and g ( x ) = x 3 3 .
Then h ( g ( x ) ) = ( x 3 3 ) 2 equals x 6 9 , not x 6 3 .
Step 2
That identity you have there is untrue, but what you may be thinking of is the chain rule which states:
d / d x ( h ( g ( x ) ) = g ( x ) h ( g ( x ) )
Spencer Lutz

Spencer Lutz

Beginner2022-05-12Added 6 answers

Explanation:
For your functions g(x), and h(x),
g ( h ( x ) ) = g ( x 2 ) = ( x 2 ) 3 3 = x 6 3 , and h ( g ( x ) ) = h ( x 3 3 ) = ( x 3 3 ) 2 = x 6 9 g ( h ( x ) ).
This means in general it does not hold that the composition of a function and its own anti-derivative commutes.

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