If f and g are monotonic, is f(g(x))?

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Homework Statement


If f and g are both increasing functions, is it true that f(g(x)) is also increasing? Either prove that it is true or five an example that proves it false.


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The Attempt at a Solution


I know that it is indeed also increasing, but I'm unsure how to prove it.
 

Answers and Replies

  • #2
Office_Shredder
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It might help to write down the definition of an increasing function
 
  • #3
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A function f(x) is increasing at a point x0 if and only if there exists some interval containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in the interval to the right of x0.
 
  • #4
D H
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A function f(x) is increasing at a point x0 if and only if there exists some interval containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in the interval to the right of x0.
How would you express that in terms of calculus?
 
  • #5
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Um idk by saying that as x increases, so does the y values.
 
  • #6
D H
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Try it without words. Write an equation. Using calculus.
 
  • #7
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If x>y then f(x)>f(y)?
 
  • #8
D H
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You posted this question in "Calculus and beyond," so use calculus.
 
  • #9
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You don't really need calculus here.

Just use that

[tex]x<y~\Rightarrow~f(x)<f(y)[/tex]

and

[tex]x<y~\Rightarrow~g(x)<g(y)[/tex]

Put these two things together.
 

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