Even, Odd or Neither Functions

mathxyz · 2024-03-20 17:15:51

Determine algebraically if each function is even, odd, or neither.

1. F(x) = x^(1/3)

Let me see.

Function F is the cube root of x.

Let cr = cube root

F(-x) = cr{-x}

F(-x) = -x

I say neither. You say?

2. h(x) = -x^3/(3x^2 - 9)

h(-x) = -(-x)^3/(3(-x)^2 - 9)

h(-x) = -(-x)^3/(3x^2 - 9)

h(-x) = x^3/(3x^2 - 9)

I say odd.

What do you say?

Bob · 2024-03-20 20:53:55

2 is good.

Here's a table of values for cube root.

x 1 8 27 -1 -8 -27

x^(1/3) 1 2 3 -1 -2 -3

What does that suggest?

Bob

ps. The MIF plotter doesn't show any negative values for x, but, as you can see, they certainly exist.

amnkb · 2024-03-21 02:06:35

FelizNYC wrote:

what happend to the cube rt?

mathxyz · 2024-03-21 02:44:16

Bob wrote:

2 is good.
Here's a table of values for cube root.
x 1 8 27 -1 -8 -27
x^(1/3) 1 2 3 -1 -2 -3
What does that suggest?
Bob
ps. The MIF plotter doesn't show any negative values for x, but, as you can see, they certainly exist.

The table suggests that the cube root of negative number is negative and the cube root of positive numbers is positive. My answer is neither. You say?

Are you saying that the cuberoot{-x} is not -x?

What is the answer? Even if the correct answer involves a complex number, for the purpose of this chapter and section in the textbook, the answer is neither odd or even.

You say?

Last edited by mathxyz (2024-03-21 02:51:23)

mathxyz · 2024-03-21 02:50:40

amnkb wrote:

FelizNYC wrote:
what happend to the cube rt?

Are you saying that the cuberoot{-x} is not -x?

What is the answer? Even if the correct answer involves a complex number, for the purpose of this chapter and section in the textbook, the answer is neither odd or even.

You say?

Bob · 2024-03-21 04:04:31

It is definitely odd.

Let's say x = P^3 = P.P.P for some real number P > 0

Then f(x) = f(P^3) = (P.P.P)^(1/3) = P

f(-x) = f(-P^3) = [(-P).(-P).(-P)]^(1/3) = -P {ie what number multiplied by itself three times makes this}

So (-P^3)^(1/3) = -P = -f(x)

Another way to check these is to look at the graph.

All even functions have the y axis as a line of symmetry.

All odd functions have rotational symmetry around the origin, order 2 (ie. same shape if rotated 180)

x^2, x^4, x^6 and so on, plus any function consisting of only even powers of x, are even functions. That's where the name comes from.

Similarly x, x^3, x^5 etc are all odd functions.

But other functions may also be even or odd as these questions show.

Bob

mathxyz · 2024-03-21 09:09:26

Bob wrote:

It is definitely odd.
Let's say x = P^3 = P.P.P for some real number P > 0
Then f(x) = f(P^3) = (P.P.P)^(1/3) = P
f(-x) = f(-P^3) = [(-P).(-P).(-P)]^(1/3) = -P {ie what number multiplied by itself three times makes this}
So (-P^3)^(1/3) = -P = -f(x)
Another way to check these is to look at the graph.
All even functions have the y axis as a line of symmetry.
All odd functions have rotational symmetry around the origin, order 2 (ie. same shape if rotated 180)
x^2, x^4, x^6 and so on, plus any function consisting of only even powers of x, are even functions. That's where the name comes from.
Similarly x, x^3, x^5 etc are all odd functions.
But other functions may also be even or odd as these questions show.
Bob

A wonderful, detailed reply. Let's hope I find time later to post a few more math problems. I start my new job on Monday, which means I will have little time to enjoy myself.

amnkb · 2024-03-22 07:13:28

FelizNYC wrote:

Are you saying that the cuberoot{-x} is not -x?

yes I'm saying they're not the same
eg cbrt{8} = 2, not 8
cbrt{27} = 3, not 27
cbrt{-64} = -4, not -64

mathxyz · 2024-03-22 09:19:14

amnkb wrote:

FelizNYC wrote:
Are you saying that the cuberoot{-x} is not -x?
yes I'm saying they're not the same
eg cbrt{8} = 2, not 8
cbrt{27} = 3, not 27
cbrt{-64} = -4, not -64

So, the cuberoot{-x} = surd(-x, 3) according to my calculator.

Math Is Fun Forum

#1 2024-03-20 17:15:51

Even, Odd or Neither Functions

#2 2024-03-20 20:53:55

Re: Even, Odd or Neither Functions

#3 2024-03-21 02:06:35

Re: Even, Odd or Neither Functions

#4 2024-03-21 02:44:16

Re: Even, Odd or Neither Functions

#5 2024-03-21 02:50:40

Re: Even, Odd or Neither Functions

#6 2024-03-21 04:04:31

Re: Even, Odd or Neither Functions

#7 2024-03-21 09:09:26

Re: Even, Odd or Neither Functions

#8 2024-03-22 07:13:28

Re: Even, Odd or Neither Functions

#9 2024-03-22 09:19:14

Re: Even, Odd or Neither Functions

Board footer