The graph of a function in practice

The following exercises are solved in this step.

We invite you to try to solve them before watching the video.

In any case, you will find below a PDF file with the solutions.

Exercise 1.

Prove that the functions [f:xmapsto x^3,quad g:xmapsto 2x^3,quad h:xmapsto frac 12 x^3,] are strictly increasing and odd, and the functions [f’:xmapsto -x^3,quad g’:xmapsto -2x^3,quad h’:xmapsto -frac 12 x^3] are strictly decreasing and odd.

Exercise 2. [Solved only in the PDF file]

Prove that the functions [f:xmapsto x^3+2,quad g:xmapsto x^3-2,quad h:xmapsto (x+2)^3,quad i:xmapsto (x-2)^3] are strictly increasing. Are they odd?