Question

Consider the domains of the expressions \(\sqrt[3]{x^{2}-7 x+12}\) and \(\sqrt{x^{2}-7 x+12}\). Explain why the domain of \(\sqrt[3]{x^{2}-7 x+12}\) consists of all real numbers.

Short answer

The domain of \(\sqrt[3]{x^{2}-7 x+12}\) is all real numbers because the cubic root of any real number is also a real number. The domain of \(\sqrt{x^{2}-7 x+12}\) is \(x \leq 3\) or \(x \geq 4\) because the square root of a number is only defined for non-negative values.

Step-by-step solution

Determine the Domain of \(\sqrt[3]{x^{2}-7 x+12}\)

The expression inside the cubic root can take any real number. This means it can be positive, zero, or negative. Therefore, the domain of \(\sqrt[3]{x^{2}-7 x+12}\) is all real numbers.

Determine the Domain of \(\sqrt{x^{2}-7 x+12}\)

The expression under the square root, \(x^{2}-7 x+12\), must be greater than or equal to zero, that is, \(x^{2}-7 x+12 \geq 0\), as square root of a negative number is not defined in the real numbers. To find the range of \(x\), we first find the roots of the equation \(x^{2}-7x+12=0\) which will give \(x=3\) or \(x=4\). Therefore, domain of \(\sqrt{x^{2}-7 x+12}\) is \(x \leq 3\) or \(x \geq 4\).

Explaining the domain of \(\sqrt[3]{x^{2}-7 x+12}\)

The cubic root of a number is defined for all real number inputs including negatives, unlike the square root which is only defined for non-negative inputs. So, the function \(\sqrt[3]{x^{2}-7 x+12}\) can take any real number and return a real number, hence the domain is all real numbers.

Key concepts

Real Numbers

Real numbers are the foundation of algebra and encompass the entire number line. This includes all the numbers we commonly use, such as rational numbers (which can be expressed as fractions) and irrational numbers (which cannot be expressed as simple fractions). Examples of rational numbers include integers like -2, 0, and 7, while examples of irrational numbers include \( \sqrt{2} \) or \( \pi \). Importantly for our discussion on domains, real numbers can be negative, positive, or zero.

In the context of finding the domain of radical expressions, the type of numbers that can be put into the expression will often determine the domain. For cubic roots, any real number is a valid input because the cubic root of a real number always produces a real number, even if the original number is negative. For square roots, however, only non-negative real numbers are valid inputs, because the square root of a negative number is not a real number, but an imaginary one.

Cubic Root

The cubic root of a number \(x\) is the number that, when multiplied by itself three times, gives \(x\). Symbolically, it is expressed as \(\sqrt[3]{x}\). Unlike square roots, cubic roots embrace the entire real number spectrum because the result of a cube can be negative, zero, or positive.

Real-World Examples

• Finding the original dimensions of a cube when you know its volume.
• Understanding the growth rate of a three-dimensional space, like a tumor, from its volume measurements.

In case of the expression \(\sqrt[3]{x^{2}-7 x+12}\), we can apply a cubic root to any output of \(x^{2}-7 x+12\), regardless of whether it's positive or negative. As such, the domain includes all real numbers.

Square Root

In sharp contrast to cubic roots, the square root of a number is the value that, when multiplied by itself, yields the original number. Mathematically, the square root of \(x\) is shown as \(\sqrt{x}\). This operation has a significant restriction; it is only defined for non-negative numbers, specifically zero and positive real numbers.

Consequences for Domain

• For an expression under a square root to have real solutions, the expression must be greater than or equal to zero.
• Expressions resulting in negative numbers are not permissible within the real number system.

As for our exercise, the expression \(\sqrt{x^{2}-7 x+12}\) can only possess real values if \(x^{2}-7 x+12 \geq 0\), limiting its domain to specific ranges of \(x\).

Inequalities

Inequalities are mathematical expressions that demonstrate the relative size or order of two values. They're symbolized by signs like \( > \), \( < \), \( \geq \), and \( \leq \), meaning greater than, less than, greater than or equal to, and less than or equal to, respectively.

When working with square roots, inequalities are vital for determining domains, as we need to ensure that the expression under the square root is non-negative. Solving the inequality \(x^{2}-7 x+12 \geq 0\) involves finding values of \(x\) that make the expression true. These values constitute the permissible domain for the square root function. We approach this problem by solving quadratic equations and factoring, leading us to the permissible ranges for \(x\), thereby establishing the domain of the function.