Domain of Function Involving Roots

Click For Summary

Discussion Overview

The discussion revolves around determining the domain of various functions involving roots, specifically cube roots, fifth roots, and sixth roots. Participants explore the implications of the type of root (even or odd) on the domain, addressing both theoretical and practical aspects of these functions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the domain of cube roots and fifth roots is all real numbers due to the nature of odd roots.
  • Others argue that for even roots, such as sixth roots, the radicand must be greater than or equal to zero to determine the domain.
  • A participant questions whether the condition of setting the radicand to be greater than or equal to zero applies to cube roots, leading to clarification that it does not.
  • Another participant provides a general rule regarding the domains of nth roots, stating that if n is even, the radicand must be non-negative, while if n is odd, any real number is acceptable.
  • Some participants discuss the implications of these rules and whether they can be applied to their specific domain questions, with some affirming that they can as long as the radical is not in a denominator.

Areas of Agreement / Disagreement

Participants generally agree that odd roots allow for all real numbers in the domain, while there is contention regarding the treatment of even roots, particularly in the context of specific functions. The discussion remains unresolved regarding the application of certain rules to all cases.

Contextual Notes

Some participants express uncertainty about the application of rules for determining domains, particularly in relation to the presence of radicals in denominators and the specific functions being discussed.

mathdad
Messages
1,280
Reaction score
0
Section 3.1
Question 2dFind the domain of the function.

Let CR = CUBE ROOT

s = CR{3t + 12}

Because it is a cube root, I say the domain is ALL REAL NUMBERS.
 
Last edited:
Mathematics news on Phys.org
Domain of Function...2

Set 3.1
Question 6d

Let CR = cube root

K(x) = CR{2x^2 + x - 6}.

Because it is a cube root, I say the domain is ALL REAL NUMBERS.
 
Last edited:
Domain of Function...3

Set 3.1
Question 10b

Let FR = fifth root

z = FR{5(x - 4)(5 - x)(x + 1)}

Because the index is an odd number, the domain is ALL REAL NUMBERS.
 
Last edited:
Domain of Function...4

Set 3.1
Question 10d

Let SR = sixth root

z = SR{5(x - 4)(5 - x)(x + 1)}

Because the index is an even number, I must set the radicand to be > or = 0.

5(x - 4)(5 - x)(x + 1) ≥ 0

Is this correct so far?
 
Last edited:
Re: Domain of Function...1

RTCNTC said:
Section 3.1
Question 2dFind the domain of the function.

Let CR = CUBE ROOT

s = CR{3t + 12}

Do I set 3t + 12 to be > or = 0? I don't recall if this applies to cube roots.

you can take cube root of -ve nuimbers also so $ - \infty <3t+12 < \infty$ or $ - \infty < t < \infty$
 
Re: Domain of Function...1

What is the cube root of, say, -8? What does the answer to this question tell you about the cube root function?
 
Re: Domain of Function...4

$$x^6=(x^2)^3$$

What can you conclude from the above? In general, what about odd powers? What about even powers?
 
Re: Domain of Function...2

RTCNTC said:
Set 3.1
Question 6d

Let CR = cube root

K(x) = CR{2x^2 + x - 6}.

Must I set the radicand to be > or = 0?

No.
 
Re: Domain of Function...3

RTCNTC said:
Set 3.1
Question 10b

Let FR = fifth root

z = FR{5(x - 4)(5 - x)(x + 1)}

Must I set the radicand to be > or = 0?

No.
 
  • #10
Re: Domain of Function...3

Consider the distinction between even roots (square roots, fourth roots, etc.) and odd roots (cube root, fourth root, etc.)
 
  • #11
Re: Domain of Function...1

I've merged the four threads pertaining to the domain of a function into one thread to facilitate a more focused discussion.
 
  • #12
Re: Domain of Function...1

greg1313 said:
What is the cube root of, say, -8? What does the answer to this question tell you about the cube root function?

The cube root of -8 is -2.

- - - Updated - - -

greg1313 said:
I've merged the four threads pertaining to the domain of a function into one thread to facilitate a more focused discussion.

That's perfectly ok. I made a mistake and posted all domain of a function questions in the Pre-University Math forum. Obviously, all my questions come from David Cohen's precalculus textbook.
 
  • #13
I found the following rules online:

f(x) = nthroot{x}

If n is even, x must be > or = 0.

Thus, f(x) is also > or = 0.

If n is odd, x can be any real number.

Thus, f(x) is also any real number.

Can I apply the above rules to my domain questions?
 
  • #14
Re: Domain of Function...1

RTCNTC said:
...I made a mistake and posted all domain of a function questions in the Pre-University Math forum. Obviously, all my questions come from David Cohen's precalculus textbook.

Even though Pre-Calculus is taught in universities, it's really considered a remedial course at that level, and so that's why we classify it as a Pre-University level course. :D
 
  • #15
Re: Domain of Function...4

kaliprasad said:
you can take cube root of -ve nuimbers also so $ - \infty <3t+12 < \infty$ or $ - \infty < t < \infty$

Sorry but I do not understand your reply.
 
  • #16
Re: Domain of Function...1

MarkFL said:
Even though Pre-Calculus is taught in universities, it's really considered a remedial course at that level, and so that's why we classify it as a Pre-University level course. :D

You are right. In fact, Calculus 1 is considered to be the first math course for people majoring in math in most CUNY colleges. When I took precalculus at Lehman College, the professor made it clear that the material in precalculus courses is taught in 11th or 12th grade in most public high schools.

Mark,

I found the following rules online:

f(x) = nthroot{x}

If n is even, x must be > or = 0.

Thus, f(x) is also > or = 0.

If n is odd, x can be any real number.

Thus, f(x) is also any real number.

Can I apply the above rules to my domain questions?
 
  • #17
Re: Domain of Function...1

RTCNTC said:
...Mark,

I found the following rules online:

f(x) = nthroot{x}

If n is even, x must be > or = 0.

Thus, f(x) is also > or = 0.

If n is odd, x can be any real number.

Thus, f(x) is also any real number.

Can I apply the above rules to my domain questions?

As long as the radical is not a factor in a denominator (in that case the radicand cannot be zero), then yes that rule can be applied for domain questions. It comes from the fact that an even power of a negative value will be positive, while an odd power of a negative value will be negative.
 
  • #18
Re: Domain of Function...1

MarkFL said:
As long as the radical is not a factor in a denominator (in that case the radicand cannot be zero), then yes that rule can be applied for domain questions. It comes from the fact that an even power of a negative value will be positive, while an odd power of a negative value will be negative.
Section 3.1
Question 2d

Find the domain of the function.

Let CR = CUBE ROOT

s = CR{3t + 12}

Because it is a cube root, I say the domain is ALL REAL NUMBERS.

Set 3.1
Question 6d

Let CR = cube root

K(x) = CR{2x^2 + x - 6}.

Because it is a cube root, I say the domain is ALL REAL NUMBERS.

Set 3.1
Question 10b

Let FR = fifth root

z = FR{5(x - 4)(5 - x)(x + 1)}

Because the index is an odd number, the domain is ALL REAL NUMBERS.

Set 3.1
Question 10d

Let SR = sixth root

z = SR{5(x - 4)(5 - x)(x + 1)}

Because the index is an even number, I must set the radicand to be > or = 0.

5(x - 4)(5 - x)(x + 1) ≥ 0

Is this correct so far?
 

Similar threads

MHB What is the domain for ax^(1/3) + b?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Transformation of Functions: How Do Domain and Range Change?
  • · Replies 4 ·
Replies
4
Views
2K
MHB Why is the domain of ax^(1/3) + b equal to all real numbers?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
3K
High School Can Negative Numbers Be in the Domain of the Square Root Function?
  • · Replies 4 ·
Replies
4
Views
5K
MHB How to Learn Domain,Range and Real Number Of Function Mathematics ?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Quadratic equations intersaction point is minimum instead of roots
  • · Replies 2 ·
Replies
2
Views
1K
MHB Domain of Rational Function: What is the domain of the given rational function?
  • · Replies 16 ·
Replies
16
Views
3K
High School Inverse Functions vs Inverse Relations
  • · Replies 4 ·
Replies
4
Views
4K
MHB Max Value of $b$ for Real Roots of $f(x)$ and $g(x)$
  • · Replies 2 ·
Replies
2
Views
1K