Skip to 0 minutes and 11 secondsHello. Welcome to the exercise of the graph of a function practice step. We have to prove that the functions f x to x cubed, g x to 2 times x cubed, h x to 1 over 2 x cubed, are strictly increasing and odd, while the functions f prime x to minus x cubed, g prime x to minus 2 times x cubed, and h prime x to minus 1 over 2 x cubed are strictly decreasing and odd. You have seen in the Francis video that the function f, which sends x to x cubed, is strictly increasing.

Skip to 1 minute and 16 secondsThat is, if x is less than y, we have that x cubed is less than y cubed. Now, let us consider two cases.

Skip to 1 minute and 38 secondsIf a is any real number greater than 0 and x is less than y, then we know that x cubed is less than y cubed. And multiplying on both sides by the positive real number a, we get that a x cubed is less than a y cubed. Therefore, considering a equal to 2 or to 1 over 2, we get that both g and h are strictly increasing.

Skip to 2 minutes and 37 secondsOn the other hand, if a is a real number less than 0, then what we get if we will repeat the similar reasoning, we get x less than y implies x cubed less than y cubed. But now multiplying by a negative number on both sides, this equality reverses, and we get a x cubed greater than a y cubed. Therefore, considering a equal to minus 1 or minus 2 or minus 1 over 2, we get that the functions f prime, g prime, and h prime are strictly decreasing.

Skip to 3 minutes and 42 secondsAnd now let us examine if these functions are odd or not. What means to be an odd function. We have for f, g, and h, we have to check which is the image of minus x. Precisely, f of minus x is equal to minus x to the cube, which is equal to minus x to the cube, which is equal to minus f of x. Therefore, the function f is odd. Similarly, g of minus x is equal to 2 times minus x to the cube, which is equal to minus 2 x to the cube, which is equal to minus g of x. And then also g is odd.

Skip to 4 minutes and 50 secondsNow, h of minus x is equal to 1 over 2 minus x to the cube, which is equal to minus 1 over 2 x cubed, which is minus h of x. And also h is an odd function. What about now f prime, g prime, and h prime? Let us compare, again, f prime of minus x with minus f prime of x and the same for the others. Then we have f prime of minus x is equal to minus minus x to the cube, which is equal to x to the cube. And what is minus f prime of x? This is equal to minus-- f prime of x is minus x cubed, which is equal to x cubed.

Skip to 5 minutes and 55 secondsTherefore, these two are equal, and then f prime is an odd function.

Skip to 6 minutes and 5 secondsNow, g prime of minus x is equal to minus 2 minus x to the cube, which is equal 2 times x cubed. And minus g prime of x is equal to minus minus 2x cubed, which is equal to 2x cubed. Again, g prime of 1 minus x is equal to minus g prime of x. And then g prime is an odd function. Finally, h prime of minus x is equal to minus 1 over 2 minus x to the cube, which is equal to 1 over 2 x to the cube. And minus h prime of x is equal to minus minus 1 over 2 x cubed, which is equal to 1 over 2 x cubed.

Skip to 7 minutes and 15 secondsAgain, h prime of minus x and minus h prime of x coincide and therefore, also h prime is an odd function.

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

Exercise 2. [Solved only in the PDF file]

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

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Precalculus: the Mathematics of Numbers, Functions and Equations

University of Padova

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