Not monotonic with increase/decrease intervals

[peripatein]

Hello,
Would it be correct to say that the function y=|(x^3)-1| is not monotonic, yet increases for every x>0 and decreases for every x<0?
I hope one of you could comment. Thanks!

 

[HallsofIvy]

I'm not sure why you use the word "yet". Any function that "increases for every x>0 and decreases for every x<0" is NOT monotonic, by definition of "monotonic". It is true that this function, because it "increases for every x>0 and decreases for every x<0" is monotonic on x> 0 and on x< 0. But not "monotonic" for all x.

 

[peripatein]

Thank you, HallsofIvy, that was certainly helpful :-).

 

[Michael Redei]

Would it be correct to say that the function y=|(x^3)-1| is not monotonic

Yes, but x=0 isn't the point at which decreasing turns into rising.

Try sketching the graph of f(x)=x³, then of g(x)=x³-1, and then of h(x)=|x³-1|. A rough sketch will do, as long as you make sure you don't add "bumps" which would change the monotonicity.