Problem 48 - Algebra and Trigonometry

Question

Find the domain of the function. $$ g(x)=\sqrt{7-3 x} $$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, \frac{7}{3}]\).

Step 1: Understanding the Function

The given function is a square root function, which is defined as \( g(x) = \sqrt{7 - 3x} \). A square root function is defined only when the expression under the square root is non-negative.

Step 2: Setting Up the Inequality

To find the domain of \( g(x) \), determine when the expression under the square root is greater than or equal to zero: \( 7 - 3x \geq 0 \).

Step 3: Solving the Inequality

Solve the inequality \( 7 - 3x \geq 0 \). Start by subtracting 7 from both sides to get \( -3x \geq -7 \). Then, divide both sides by -3, reversing the inequality sign, to get \( x \leq \frac{7}{3} \).

Step 4: Expressing the Domain

The solution to the inequality \( x \leq \frac{7}{3} \) describes the domain of the function. Therefore, the domain of \( g(x) \) is all real numbers \( x \) such that \( x \leq \frac{7}{3} \). In interval notation, this is expressed as \( (-\infty, \frac{7}{3}] \).

Key Concepts

Square Root Function

Square root functions are a specific type of function involving the square root of an expression. The general form of a square root function is \( f(x) = \sqrt{a} \), where \( a \) is called the radicand. The primary rule governing these functions is that the radicand must be non-negative to ensure the function is defined in the real number system.

• A positive radicand allows a real number output, while a negative radicand would yield an imaginary number result, not considered in its domain within real numbers.
• This constraint informs us to focus on solving inequality problems where a \( \geq 0 \) (greater than or equal to zero) is aimed.

In our exercise, the function \( g(x) = \sqrt{7 - 3x} \) has a radicand of \( 7 - 3x \). Our goal is to find values of \( x \) that keep \( 7 - 3x \) non-negative, ensuring the resulting function value is a real number.

Inequality Solving

Solving inequalities is crucial when finding the domain of functions involving square roots. Inequalities help identify the range of values where the function remains real. Here's a simple guide on how we solve the inequality step by step.

• Begin with setting up an inequality based on the requirement that the radicand must be non-negative, for example, \( 7 - 3x \ge 0 \).
• To solve, isolate \( x \) by performing arithmetic operations until \( x \) is on one side of the inequality.

In our step-through:
- Subtract 7 from both sides to get \( -3x \ge -7 \). Remember!- When dividing by a negative number, the inequality sign reverses, making \( x \le \frac{7}{3} \).
This solution tells us the allowable values for \( x \) that make the entire expression under the square root non-negative.

Interval Notation

Interval notation is a concise way to express the domain or range of a function. It represents all the possible values a function can take under certain conditions and involves brackets and parentheses. Understanding how to interpret interval notation is key:

• Parentheses \(( )\) denote that an endpoint is not included in the interval.
• Brackets \([ ]\) indicate the endpoint is included.
• An interval like \((-\infty, \frac{7}{3}]\) defines all numbers less than or equal to \( \frac{7}{3} \).

For the function \( g(x) = \sqrt{7 - 3x} \), the domain where the function output remains real is expressed in interval notation as \( (-\infty, \frac{7}{3}] \). This indicates that \( x \) can be any real number up to and including \( \frac{7}{3} \), starting from negative infinity.