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Chapter 10: Problem 98

Determine the domain of each function described. $$ g(t)=\sqrt[3]{2 t-6} $$

Short Answer

Expert verified
The domain of \( g(t) = \sqrt[3]{2t - 6} \) is all real numbers, \(-\infty, \infty\).

Step by step solution

01

Understanding the Function

The given function is a cube root function: \( g(t) = \sqrt[3]{2t - 6} \). We need to determine the domain of this function.
02

Identify Domain Constraints

Unlike square root functions, cube root functions are defined for all real numbers because the cube root of any real number exists. Thus, there are no constraints on the expression inside the cube root, \( 2t - 6 \).
03

State the Domain

Since there are no constraints for the function \( g(t) = \sqrt[3]{2t - 6} \), the domain includes all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Root Function
A cube root function is a type of mathematical function represented as \( g(t) = \sqrt[3]{x} \).
The cube root, unlike the square root, can take positive, negative, and zero values as inputs and still give a real number as output.
This flexibility makes the cube root function unique compared to higher even-degree roots.
For instance:
  • \( \sqrt[3]{-8} = -2 \)
  • \( \sqrt[3]{0} = 0 \)
  • \( \sqrt[3]{27} = 3 \)
Because of this broad range, cube root functions are always defined across all real numbers.
Real Numbers
Real numbers include both rational and irrational numbers. Rational numbers can be expressed as fractions while irrational numbers cannot be precisely written as fractional expressions.
Examples of real numbers:
  • Positive numbers: 5, 3.14, 2
  • Negative numbers: -1, -0.75, -4
  • Zero: 0
This complete set of real numbers ensures that functions like cube root functions remain fully applicable across an infinite range of values without restrictions.
For the given function \( g(t) = \sqrt[3]{2t - 6} \), any value of \( t \) within the real number set enables a valid output.
Function Constraints
Function constraints are conditions or limits placed on the domain of a function. Some functions, like square roots or fractional functions, have constraints to ensure their outputs remain defined and real.
Examples of function constraints:
  • Square root function \( \sqrt{x} \: x \geq 0 \)
  • Rational function \( \frac{1}{x}\: x \eq 0 \)
However, for the cube root function \( g(t) = \sqrt[3]{2t - 6} \) there are no such constraints.
The expression inside the cube root, \( 2t - 6 \), does not limit the domain, since cube roots exist for all real numbers.
Hence, the domain of \( g(t) = \sqrt[3]{2t - 6} \) includes all real numbers, with no constraints.

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