Problem 3 - College Algebra

Question

Explain why the domain of \(f(x)=\sqrt[3]{x}\) is different from the domain of \(f(x)=\sqrt{x}\)

Step-by-Step Solution

Answer

The domain of \(\sqrt[3]{x}\) is all real numbers, while \(\sqrt{x}\) only accepts non-negative numbers.

Step 1: Understanding the Domain of a Function

The domain of a function is the set of all possible input values (typically x-values) for which the function is defined. For a function to be defined at a certain point, it must yield a real number as an output.

Step 2: Analyze the Cube Root Function

Consider the function \(f(x) = \sqrt[3]{x}\). The cube root of any real number is defined. Thus, \(\sqrt[3]{x}\) can accept any real number as input, making its domain all real numbers \((-\infty, \infty)\).

Step 3: Analyze the Square Root Function

Consider the function \(f(x) = \sqrt{x}\). The square root of a negative number is not a real number, so \(\sqrt{x}\) can only accept non-negative numbers as input. Therefore, its domain is \([0, \infty)\).

Step 4: Compare the Domains

The domain of \(f(x) = \sqrt[3]{x}\) includes all real numbers, unlike \(f(x) = \sqrt{x}\), which is restricted to non-negative numbers. This difference arises because the cube root can handle negatives, whereas the square root cannot.

Key Concepts

Cube Root Function

The cube root function, denoted in mathematics as \(f(x) = \sqrt[3]{x}\), takes any real number as input. This is because the domain of a cube root function encompasses all real numbers \(( -\infty, \infty )\). This means you can plug in any positive, negative, or zero value for \(x\) and the function will yield a real number output.

The reason behind this ability lies in the nature of exponents. In simpler terms, for any real number \(x\), the cube root provides a real number that, when raised to the power of three, gives back \(x\).

• For instance, \(\sqrt[3]{8} = 2\), because \(2^3 = 8\).
• Similarly, \(\sqrt[3]{-8} = -2\), as \((-2)^3 = -8\).

Thus, cube root functions are versatile and can handle negative inputs seamlessly.

Square Root Function

Unlike the cube root, the square root function \(f(x) = \sqrt{x}\) is only defined for non-negative real numbers. This means the domain of this function is restricted to the set of numbers \([0, \infty)\).

The restriction arises because you cannot take the square root of a negative number and end up with a real number. In simple terms, there is no real number that squares to result in a negative number. Hence, only non-negative inputs yield real number outputs with this function.

• For example, \(\sqrt{4} = 2\), since \(2^2 = 4\).
• However, \(\sqrt{-4}\) does not result in a real number within the realm of real numbers, as no square of a real number equals \(-4\).

The domain's limitation makes square root functions less flexible compared to cube root functions.

Real Numbers

Real numbers are a set of numbers that include all the rational and irrational numbers. They form a continuum of values that don't exclude any positive, negative, or fractional values.

• Rational numbers are numbers that can be expressed as a fraction, such as \(\frac{1}{2}\) or -5.
• Irrational numbers cannot be written as simple fractions, like \(\pi\) or \(\sqrt{2}\).

Real numbers play a crucial role in defining domains of functions, as the functions need to yield real number outputs to be considered defined for certain inputs.

In both cube and square root functions, determining when they produce real number outputs is essential for identifying their domains. While the cube root function accepts all real number inputs (including negatives) and outputs real numbers, the square root is limited to non-negative inputs to ensure the output remains a real number.

Hence, understanding real numbers helps in appreciating why certain functions have the domains that they do.