Problem 20 - Calculus with Concepts in Calculus

Question

Find the domain of the function. $$ f(x)=\sqrt{2-3 x} $$

Step-by-Step Solution

Answer

The domain is \((-\infty, \frac{2}{3}]\).

Step 1: Understanding the Function

We are given the function \( f(x) = \sqrt{2 - 3x} \). This is a square root function, and the expression inside the square root, \( 2 - 3x \), must be non-negative for the square root to be defined. Therefore, we will find the domain by determining where \( 2 - 3x \geq 0 \).

Step 2: Setting Up the Inequality

To find the domain, set up the inequality for the expression inside the square root. We have:\[ 2 - 3x \geq 0 \].

Step 3: Solving the Inequality

Now, solve the inequality \( 2 - 3x \geq 0 \).Start by isolating \( x \):1. Subtract 2 from both sides: \( -3x \geq -2 \).2. Divide by \(-3\) and reverse the inequality sign (since we are dividing by a negative number):\[ x \leq \frac{2}{3} \].

Step 4: Expressing the Domain

The solution to the inequality \( x \leq \frac{2}{3} \) gives us the domain of the function. This means \( f(x) \) is defined for all values of \( x \) that are less than or equal to \( \frac{2}{3} \). In interval notation, the domain is \( (-\infty, \frac{2}{3}] \).

Key Concepts

Square Root Function

A square root function is a type of mathematical function that involves the square root of an expression. In the given function, \( f(x) = \sqrt{2 - 3x} \), the square root symbol indicates that we are looking for a value that, when squared, yields the expression inside.

For the square root to be real and defined, the expression inside the root, \( 2 - 3x \), must be greater than or equal to zero. This condition ensures that we do not encounter imaginary numbers, as the square root of a negative number is not defined in the real number system.

It's crucial to identify this condition when determining the domain of a square root function because it outlines the range of \( x \)-values that will yield real outputs. If you pay attention to these details, solving for the domain becomes straightforward.

Inequality Solving

Solving inequalities is an important step in finding the domain of functions like the square root function. Let's continue with the example of \( f(x) = \sqrt{2 - 3x} \), where we need to solve \( 2 - 3x \geq 0 \) to ensure the expression is non-negative.

Here's a step-by-step process:

• Subtract 2 from both sides: \(-3x \geq -2\).
• Divide each side of the inequality by \(-3\), remembering to reverse the inequality symbol, which results in \(x \leq \frac{2}{3}\).

This method lets us determine the range of \( x \)-values that keep the function real. It's essential to consider the rules of inequalities, especially the one that mandates flipping the inequality sign when multiplying or dividing by a negative number.

The solution \( x \leq \frac{2}{3} \) successfully finds the domain's boundary where the square root remains valid—no tricks, just systematic solving!

Interval Notation

Interval notation provides a concise way of expressing the domain of a function. From our earlier inequality \( x \leq \frac{2}{3} \), we determine that all real numbers less than or equal to \( \frac{2}{3} \) are valid inputs for our function.

To express this domain, we use interval notation which is written as \(( -\infty, \frac{2}{3}]\).

• The round bracket "(" indicates that \(-\infty\) is not included because infinity itself isn't a real number.
• The square bracket "]" shows that \(\frac{2}{3}\) is included in the domain.

This notation is efficient and widely used in mathematics to represent intervals, particularly when discussing domains and ranges of functions.

Understanding interval notation makes it simpler for students to interpret solutions and communicate mathematical concepts clearly.