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nycguitarguy

Member

Registered: 2024-02-24

Posts: 545

Even, Odd or Neither Functions

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

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Even, Odd or Neither Functions

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.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

amnkb

Member

Registered: 2023-09-19

Posts: 253

Re: Even, Odd or Neither Functions

what happend to the cube rt?

nycguitarguy

Member

Registered: 2024-02-24

Posts: 545

Re: Even, Odd or Neither Functions

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 nycguitarguy (2024-03-21 02:51:23)

nycguitarguy

Member

Registered: 2024-02-24

Posts: 545

Re: Even, Odd or Neither Functions

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

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Even, Odd or Neither Functions

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

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 545

Re: Even, Odd or Neither Functions

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

Member

Registered: 2023-09-19

Posts: 253

Re: Even, Odd or Neither Functions

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

nycguitarguy

Member

Registered: 2024-02-24

Posts: 545

Re: Even, Odd or Neither Functions

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.