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For functions $f,g: \mathbb{R} \to \mathbb{R}$ prove the following:

1) If $f$ and $g$ are monotonic going up so is $f+g$

2) if $f$ and $g$ are monotonic going up so is $f \cdot g$

3) if $f$ and $g$ are monotonic going up so is $f \circ g$

I know for sure that 1 is possible and 2 is not. I am not sure for 3, I am stuck on how to write down the proof.

Thanks in advance

mookid
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Ben Porat
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1 Answers1

2

1) You just need to write the definition of monotonic and going up

2) Consider the following functions :

$$f(x)=x-1$$

$$g(x)=x$$

Their product is $x^2-x$ and is not monotonic (changes in $x=1/2$)

3) The definition of "g is monotonic and going up" is the following :

$$x<y => g(x)<g(y)$$

so if you say $X = g(x)$ and $Y = g(y)$ you have :

$$X < Y$$

It means you can apply the definition for f this time :

$$f(X) < f(Y)$$

And if you switch back to x and y, you have :

$$f(g(x))<f(g(y))$$

This is true for any x and y, it means that $f \circ g$ is monotonic going up

T_O
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