Problem 22 - Elementary and Intermediate Algebra

Question

Translate each sentence into mathematical symbols. a. \(f\) of \(x\) equals the square root of the quantity \(x\) minus five. b. \(g\) of \(x\) equals the cube root of \(x\) squared.

Step-by-Step Solution

Answer

a. \(f(x) = \sqrt{x - 5}\); b. \(g(x) = \sqrt[3]{x^2}\).

Step 1: Understanding the Problem (a)

We need to express the sentence 'f of x equals the square root of the quantity x minus five' using mathematical symbols. The term 'f of x' can be written as \(f(x)\). The phrase 'the square root' is represented by the square root symbol \(\sqrt{}\). The remaining part 'of the quantity x minus five' indicates our expression inside the square root, \(x - 5\).

Step 2: Formulating Solution (a)

Combine the elements identified: \(f(x)\) is the expression, and \(\sqrt{x - 5}\) is what it equals to. So we write the equation as: \[ f(x) = \sqrt{x - 5} \].

Step 3: Understanding the Problem (b)

Here we translate 'g of x equals the cube root of x squared.' The term 'g of x' is written as \(g(x)\). The 'cube root' is symbolized by \(\sqrt[3]{}\). Finally, the 'x squared' is written as \(x^2\). These should be combined following the order mentioned.

Step 4: Formulating Solution (b)

Translate each part into symbols: \(g(x)\) and \(\sqrt[3]{x^2}\). Combine them to form the equation: \[ g(x) = \sqrt[3]{x^2} \].

Key Concepts

Square Root

The square root is a fundamental concept in mathematics, especially in the study of algebraic functions. It is commonly introduced in primary education and involves determining which number, when multiplied by itself, gives the original number under consideration. For example, the square root of 9 is 3, because 3 times 3 equals 9. Mathematically, the square root of a number \( y \) is denoted as \( \sqrt{y} \). Here are some key points about square roots:

• The square root symbol is \( \sqrt{} \), and the expression inside the symbol is known as the "radicand."
• Square roots only apply to non-negative numbers within the real number system, as negative numbers would yield an imaginary number when squared.
• In equations, square root operations are typically undone with squaring, which is the inverse operation of finding a square root.

In the context of algebraic functions, expressing sentences like "f of x equals the square root of the quantity \( x \) minus five" involves recognizing the operation that needs to be translated. Hence, you express it as \( f(x) = \sqrt{x - 5} \). This achieves a mathematical translation from words to symbols, allowing mathematicians and students to process and manipulate the equation further.

Cube Root

Understanding the cube root involves recognizing a similar principle to square roots, but adjusted to a three-dimensional context. Instead of asking what number, when multiplied by itself, equals the original number, cube roots consider which number, when cubed (multiplied by itself twice more), results in the given number. Mathematically, the cube root of a number \( z \) is denoted as \( \sqrt[3]{z} \).

• The cube root symbol is \( \sqrt[3]{} \), and like square roots, the expression inside is the radicand.
• Cube roots can have both positive and negative solutions in the real number system, unlike square roots. For example, the cube root of 8 is 2 because 2 cubed equals 8, and the cube root of -8 is -2, as \(-2\) cubed equals -8.
• Cube roots are commonly used, especially in geometry and when discussing volume or identity functions.

In our example, to translate "g of x equals the cube root of \( x \) squared" into mathematical notation, identify and assign symbols for each part of the phrase. The structure becomes \( g(x) = \sqrt[3]{x^2} \), highlighting how the algebraic operation combines evaluating the cube root of a quantity squared.

Algebraic Functions

Algebraic functions are an integral part of mathematics, forming the backbone for many advanced studies in mathematical theory. They consist of polynomial equations and operations of addition, subtraction, multiplication, division, and root extraction. An algebraic function can take a wide variety of forms but typically involves expressions where the variables are subject to algebraic operations involving coefficients and exponents.

• They can be simple, like linear functions \( f(x) = mx + b \), or more complex, involving multiple layers of operations, like rational functions and radicals.
• Algebraic functions often require translation from verbal descriptions into mathematical symbols to be analyzed or solved.
• Complex algebraic functions can illuminate relationships between quantities, model real-world phenomena, or simplify computational processes.

In translating sentences to algebraic functions, understanding each component and its association with mathematical operations is crucial. For instance, converting "f of x equals the square root of the quantity \( x \) minus five" into \( f(x) = \sqrt{x - 5} \) showcases how algebraic understanding bridges language and numeric expression, facilitating better analysis and comprehension of mathematical relationships.