Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-32: Find the domain of \(f\). $$ f(x)=\sqrt{8-3 x} $$

Step-by-Step Solution

Answer

The domain of \(f(x) = \sqrt{8-3x}\) is \((-\infty, \frac{8}{3}]\).

Step 1: Understand the Function

The function is given as \( f(x) = \sqrt{8-3x} \). This is a square root function, which implies that the expression inside the square root must be greater than or equal to zero, because the square root of a negative number is not defined in the real number system.

Step 2: Set the Inequality

To find the domain, set the inequality so that the expression inside the square root is non-negative. This means setting up the inequality: \[8 - 3x \geq 0\]

Step 3: Solve the Inequality

Solve the inequality \(8 - 3x \geq 0\): - Subtract 8 from both sides: \[ -3x \geq -8 \]- Divide both sides by \(-3\) and reverse the inequality sign (because we are dividing by a negative number): \[ x \leq \frac{8}{3} \] Thus, the inequality solution is \(x \leq \frac{8}{3}\).

Step 4: Write the Domain

The range of \(x\) values that satisfy the inequality \(x \leq \frac{8}{3}\) form the domain of the function. In interval notation, the domain of \(f(x)\) is:\[ (-\infty, \frac{8}{3}] \]

Key Concepts

Square Root Function

A square root function involves taking the square root of an expression. In its simplest form, it is written as \( f(x) = \sqrt{x} \).
The simplest square root function has a domain restriction because the square root of a negative number is not defined in the real numbers. This means that the expression inside the square root must be zero or positive, i.e., \( x \geq 0 \).
In our example, \( f(x) = \sqrt{8 - 3x} \), the expression under the square root, \( 8 - 3x \), must satisfy the same requirement: it must be greater than or equal to zero.
Therefore, solving this will help us understand where the function exists on the real number line.

Inequality Solving

Inequality solving is a significant skill when finding the domain of functions involving square roots and other similar situations.
This process involves arranging the given inequality, then simplifying it step by step.
For the function \( f(x) = \sqrt{8 - 3x} \), we set up the inequality \( 8 - 3x \geq 0 \) to determine where the expression under the square root is non-negative.
Solving inequalities follows rules similar to equations, but with a crucial difference: when you multiply or divide both sides by a negative number, the inequality sign must be reversed.
In our example, solving \( -3x \geq -8 \) involves:

• Subtracting 8 from both sides.
• Dividing by \(-3\), while reversing the \(\geq\) sign to \(x \leq \frac{8}{3}\).

This solution provides us with all \(x\) values that satisfy the original inequality.

Interval Notation

Interval notation is a concise way of writing the range of values that solve an inequality or represent a domain.
It uses brackets and parentheses:

• Square brackets \([\underline{\phantom{xxx}} ]\) indicate that the endpoint is included in the interval (i.e., the inequality has \( \leq \) or \( \geq \)).
• Parentheses \(( \underline{\phantom{xxx}} )\) indicate that the endpoint is not included \(\lt\) or \( \gt\).

In the context of our exercise, we found the domain of \( f(x) = \sqrt{8 - 3x} \) to be \( x \leq \frac{8}{3} \), meaning all real numbers less than or equal to \(\frac{8}{3}\).
Therefore, the domain in interval notation is \(( -\infty, \frac{8}{3}]\).
Infinity always gets a parenthesis because it is not an actual number, so it isn't possible to "reach" infinity.