Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Radical Expressions

College Algebra

0. Review of Algebra

Radical Expressions

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2:08 minutes

Problem 123

Textbook Question

___The domain of f(x) = ³√x−4 is [4, ∞).

Verified step by step guidance

Identify the function: \( f(x) = \sqrt[3]{x-4} \).

Understand that the cube root function \( \sqrt[3]{x} \) is defined for all real numbers.

Recognize that the expression inside the cube root, \( x-4 \), must be defined for all real numbers.

Since there are no restrictions on \( x \) for the cube root, \( x-4 \) can be any real number.

Conclude that the domain of \( f(x) = \sqrt[3]{x-4} \) is all real numbers, which is \((-\infty, \infty)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function f(x) = ³√x−4, the domain is determined by the values of x that do not lead to undefined expressions. In this case, since the cube root function is defined for all real numbers, the domain starts from 4 and extends to positive infinity.

Cube Root Function

The cube root function, denoted as ³√x, is the inverse operation of cubing a number. It takes a real number x and returns a value y such that y³ = x. Unlike square roots, cube roots can accept negative values, which means they are defined for all real numbers. This property is crucial in determining the domain of the function f(x) = ³√x−4.

Vertical Shift

A vertical shift occurs when a constant is added or subtracted from a function, affecting its output values without altering its input values. In the function f(x) = ³√x−4, the '-4' indicates a downward shift of the cube root function by 4 units. This shift does not impact the domain but changes the range of the function, which is important for understanding its overall behavior.