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Chapter 1: Problem 54

Find the domain of the function. $$f(t)=\sqrt[3]{t+4}$$

Short Answer

Expert verified
The domain of the function \(f(t)=\sqrt[3]{t+4}\) is all real numbers.

Step by step solution

01

Identify the type of function

The function \(f(t)=\sqrt[3]{t+4}\) is a cubic root function (or, more generally, a radical function). Radical functions are defined for all real values of t under the root which satisfy the inequalities defined by the radicand. For a cubic root (or any root of odd degree), all real numbers are permitted, since the cube root of both negative and positive numbers is defined.
02

Define the domain

In the case of the function \(f(t)=\sqrt[3]{t+4}\), any real number t that we choose will be a legal input because the cubic root of t+4 will always yield a real number. Since there is no restriction on the value of t that we can choose, the domain of this function is all real numbers.

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Key Concepts

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

Cubic Root Function
The cubic root function is a particular type of radical function that involves finding the cube root of the input variable. In mathematical terms, for a function such as \( f(t) = \sqrt[3]{t+4} \), the goal is to determine the cube root of the expression \( t + 4 \). Unlike square roots, which require the radicand to be non-negative for real number solutions, cubic roots can handle both negative and positive inputs. This means the cube root of a negative number exists and is also a negative number. As a result, cubic root functions are very flexible in terms of their domain.

Key characteristics of cubic root functions include:
  • Their graphs are symmetric about the origin, displaying an odd function behavior.
  • They increase continuously without bound, showing that they have no maximum or minimum values.
  • The function \( f(t) = \sqrt[3]{t} \) serves as the basic parent function for all other cubic root functions.
Radical Functions
Radical functions involve roots of variables, and their general form can be represented as \( f(x) = \sqrt[n]{g(x)} \), where \( n \) represents the degree of the root. The domain of radical functions can vary depending on the type of root involved. While square roots require the radicand to be non-negative for real numbers, odd roots like cubic roots allow the radicand to be any real number.

Important features of radical functions include:
  • For even roots (e.g., square roots), the domain is restricted to non-negative radicands because real negative results aren't in the real number set.
  • Odd roots, such as the cubic root, accept all real numbers as input, because they鈥檙e "less picky" with the input's sign.
  • The complexity of the function's graph can vary significantly, depending on the degree of the root.
Real Numbers
The concept of real numbers is a fundamental one in mathematics, encompassing all numbers that can be found on the number line. This includes integers, fractions, and irrational numbers - basically any number that isn鈥檛 imaginary. Real numbers are split into rational and irrational numbers:

  • Rational numbers are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).
  • Irrational numbers cannot be written as fractions and have non-repeating, non-terminating decimal forms, such as \( \pi \) and \( \sqrt{2} \).
In terms of functions, especially those involving radicals such as the cubic root, real numbers play a crucial role in determining the domain. Since the domain of the cubic root function is all real numbers, this means any real number can be substituted into the function, making it very versatile and comprehensive for mathematical analysis.

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Most popular questions from this chapter

(a) Given a function \(f,\) prove that \(g(x)\) is even and \(h(x)\) is odd, where \(g(x)=\frac{1}{2}[f(x)+f(-x)]\) and \(h(x)=\frac{1}{2}[f(x)-f(-x)].\) (b) Use the result of part (a) to prove that any function can be written as a sum of even and odd functions. [Hint: Add the two equations in part (a).] (c) Use the result of part (b) to write each function as a sum of even and odd functions. \(f(x)=x^{2}-2 x+1, \quad k(x)=\frac{1}{x+1}\)

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