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Chapter 7: Problem 98

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

Short Answer

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

Step by step solution

01

Analyze the function type

Observe that the function is a cube root function: \( g(t) = \sqrt[3]{2t - 6} \). Cube root functions are defined for all real numbers because the cube root of any real number is also a real number.
02

Confirm function domain

Since cube root functions do not have restrictions on the values inside the radical (unlike square roots), the expression \(2t - 6\) can take any real number without causing any issues. Therefore, there are no restrictions on the domain.
03

State the domain

Conclude that the domain of the function \( g(t) = \sqrt[3]{2t - 6} \) is all real numbers. In interval notation, this can be expressed as \((-\infty, \infty)\).

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Key Concepts

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

cube root function
The function given in the problem is a cube root function, denoted as \( g(t) = \sqrt[3]{2t - 6} \). A cube root function involves taking the cube root of an expression. Unlike square root functions, cube root functions are quite flexible. The cube root function is defined for all real numbers. This means you can plug in any real number into \( t \) and you will always get a real number out. Whether you have a positive, negative, or zero inside the radical sign, you can still find the cube root of that value.
real numbers
When we discuss whether a function is defined for 'all real numbers', it means the input can be any number from the set of real numbers. Real numbers include every number you can think of: positive numbers, negative numbers, fractions, and irrational numbers. In mathematical sets notation, the set of real numbers is often denoted by \( \mathbb{R} \). Hence, for the cube root function \( g(t) = \sqrt[3]{2t - 6} \), you can substitute any real number for \( t \). For every possible value of \( t \), \( g(t) \) will deliver a real number.
interval notation
To express the domain of a function using interval notation, we detail the set of all possible input values. For domains covering all real numbers, we use the notation \( (-\infty, \infty) \). This means the function can take any value from negative infinity to positive infinity. For the cube root function \( g(t) = \sqrt[3]{2t - 6} \), there are no restrictions on the inputs based on what we discussed earlier.
Hence, the interval notation for the function鈥檚 domain is \( (-\infty, \infty) \). This is a compact way of telling us that the function works for every real number.

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