Problem 22 - 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 Terms

We need to convert words into mathematical symbols. The phrase "\(f\) of \(x\)" is typically written as \(f(x)\). Similarly, "\(g\) of \(x\)" becomes \(g(x)\). Pay attention to operations like square root and cube root as well as arithmetic operations like addition, subtraction, and exponents.

Step 2 - Sentence A: Setting Up the Expression

For the sentence: "\(f\) of \(x\) equals the square root of the quantity \(x\) minus five," convert it as follows: Begin with \(f(x)=\) since it refers to the function value. The sentence tells us that this function value is the square root of \(x-5\). Thus, the expression becomes \(f(x) = \sqrt{x-5}\).

Step 3 - Sentence B: Setting Up the Expression

For the sentence: "\(g\) of \(x\) equals the cube root of \(x\) squared," follow similar steps. Start with \(g(x) =\) to establish the function value. Then interpret the operation: "the cube root of \(x\) squared" can be represented as \(\sqrt[3]{x^2}\). Therefore, the expression becomes \(g(x) = \sqrt[3]{x^2}\).

Step 4 - Combine the Translations

Gather the translated expressions for both sentences: (a) \(f(x) = \sqrt{x-5}\) (b) \(g(x) = \sqrt[3]{x^2}\). These expressions correctly represent the original sentences in mathematical notation.

Key Concepts

Function Notation

Function notation is a way to represent functions in mathematical expressions. The term "function" generally describes a relationship between inputs and outputs. Instead of writing out the relationship in words, we use symbols to make things easier. The notation involves using a letter, such as \( f \) or \( g \), to label the function, and \( x \) to represent the input value.
For example, in the expression \( f(x) \), \( f \) is the name of the function, and \( x \) is the variable that we input into the function. When you see \( f(x) \), it means "the value of the function \( f \) when the input is \( x \)."

• In the problem, \( f(x) \) and \( g(x) \) denote the function \( f \) and \( g \), respectively, applied to the variable \( x \).
• This helps in translating verbal expressions into accurate mathematical language.

Square Root

The square root is a concept that finds a number which, when multiplied by itself, gives the original number. It is an operation opposite to squaring a number.
Visually, it's often represented as \( \sqrt{x} \), meaning "the square root of \( x \)."

• For the given exercise, the square root function is used to express \( f(x) = \sqrt{x-5} \).
• Here, \( x-5 \) is under the square root symbol, which means it's the expression being evaluated.
• This operation conveys taking the square root of a quantity that is first adjusted by subtracting 5 from \( x \).

Cube Root

The cube root is a similar concept to the square root but involves finding a number which, when multiplied by itself three times, results in the original number. The cube root is denoted by \( \sqrt[3]{x} \).
For our exercise, understanding this operation helps translate the verbal expression into math symbols correctly.

• The expression \( g(x) = \sqrt[3]{x^2} \) applies the cube root to \( x^2 \).
• Here, it means we first square \( x \) and then take the cube root of the result.
• It's critical to perform the squaring before taking the cube root, as it shows the correct order of operations needed to reach the solution.

Translating Verbal Expressions

Translating verbal expressions into mathematical symbols involves understanding the language of math. Each word corresponds to a specific operation or relation in math. The key is to break down each component and accurately convey that into symbols.
In our case:

• The phrase "\( f \) of \( x \) equals" translates to \( f(x)= \).
• "The square root of the quantity \( x \) minus five" becomes \( \sqrt{x-5} \).
• Similarly, "\( g \) of \( x \) equals" translates to \( g(x)= \).
• "The cube root of \( x \) squared" is represented by \( \sqrt[3]{x^2} \).

This translation is crucial for converting real-world problems into a form that can be solved using mathematical tools. Understanding the correct order and the specific terminology used in math is essential for accurate expression formation.