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

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In summary, a monotonic function is one that always increases or decreases as its input increases. The monotonicity of f and g affects the composition f(g(x)), with both functions being monotonic resulting in a monotonic composition. It is not possible for f(g(x)) to be non-monotonic if f and g are both monotonic. However, it is possible for f and g to be non-monotonic and still result in a monotonic composition if the non-monotonicity of one function cancels out the other's. Finally, the behavior of the composition f(g(x)) is determined by the monotonicity of f and g, with both increasing or decreasing resulting in an increasing or decreasing composition.
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NWeid1
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1. 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.


2. Homework Equations



3. The Attempt at a Solution
I know that it is indeed also increasing, but I'm unsure how to prove it.
 
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  • #3
Thanks for the help.
 
  • #4
NWeid1 said:
Thanks for the help.

micromass's advice is quite good. Pay attention to it.
 

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

What does it mean for a function to be monotonic?

A monotonic function is one that either always increases or always decreases as its input increases. In other words, if the input value increases, the output value will also increase (or decrease) at a constant rate.

How does the monotonicity of f and g affect the composition f(g(x))?

If both f and g are monotonic, then the composition f(g(x)) will also be monotonic. This means that the output of the composite function will either always increase or always decrease, depending on the monotonicity of the individual functions.

Can f(g(x)) be not monotonic if f and g are both monotonic?

No, if both f and g are monotonic, then the composition f(g(x)) will always be monotonic. This is because the monotonicity of a function is determined by its input and output values, not by the function itself.

Can f and g be non-monotonic and still result in a monotonic composition f(g(x))?

Yes, it is possible for f and g to be non-monotonic but still result in a monotonic composition f(g(x)). This can happen if the non-monotonicity of one function cancels out the non-monotonicity of the other, resulting in a monotonic overall function.

How does the monotonicity of f and g affect the behavior of the composition f(g(x))?

The monotonicity of f and g determines the overall behavior of the composition f(g(x)). If both functions are increasing, then the composition will also be increasing. If both are decreasing, then the composition will be decreasing. However, if one is increasing and the other is decreasing, the behavior of the composition will depend on the specific values of f and g.

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