MHB What is the domain for variables with cube roots?

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The domain for the expression ax + b is all real numbers (R), as there are no restrictions on x. For the expression ax^(1/3) + b, the cube root function allows for any real number, meaning the domain is also all real numbers. Negative values of x can have real cube roots, as demonstrated by the example of (-8)^(1/3) equaling -2. Therefore, the correct domain for both expressions is R, encompassing all real numbers. The discussion clarifies that cube roots do not impose restrictions on the variable x.
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Specify the domain of each variable.

1. ax + b

Here x can be any integer.

In that case the domain is R, where R is ALL REAL NUMBERS.

2. ax^(1/3) + b

Let x^(1/3) be the cube root of x.

Let D = domain

x^(1/3) is > or = 0

[x^(1/3)]^3 > or = 0^(1/3)

x > or = 0

D = {x| x > or = 0}

Is any of this right?
 
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RTCNTC said:
Specify the domain of each variable.

1. ax + b

Here x can be any integer.

In that case the domain is R, where R is ALL REAL NUMBERS.

Correct, there would be no restriction we would need to impose on $x$.

RTCNTC said:
2. ax^(1/3) + b

Let x^(1/3) be the cube root of x.

Let D = domain

x^(1/3) is > or = 0

[x^(1/3)]^3 > or = 0^(1/3)

x > or = 0

D = {x| x > or = 0}

Is any of this right?

Let's think about this...what is the cube of a negative number?
 
Negative numbers can't have real number square roots, but negative numbers can have real number cube roots.

Sample: (-8)^(1/3) = -2

Back to my question.

(x)^(1/3) is the cube root of x. The domain can be any real number.
 
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