Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, find the domain of each function using interval notation. \(f(x)=\sqrt{4-3 x}\)

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, \frac{4}{3}]\).

Step 1: Identify the Condition for the Radicand

The function given is a square root function, \( f(x) = \sqrt{4-3x} \).For a square root to be defined, the expression inside, the radicand, \(4-3x\), must be greater than or equal to zero. Thus, we set up the inequality: \( 4-3x \geq 0 \).

Step 2: Solve the Inequality

Solve the inequality \(4-3x \geq 0\) to find the values of \(x\) for which the function is defined. - Start by subtracting 4 from both sides: \( -3x \geq -4 \). - Next, divide every term by \(-3\). Remember, dividing by a negative flips the inequality sign: \( x \leq \frac{4}{3} \). So, the solution to the inequality is \( x \leq \frac{4}{3} \).

Step 3: Express the Domain in Interval Notation

The domain of \(f(x)\) is the set of all \(x\) values that satisfy the inequality \(x \leq \frac{4}{3}\). In interval notation, this is written as \((-\infty, \frac{4}{3}]\), indicating all real numbers less than or equal to \(\frac{4}{3}\).

Key Concepts

Square root functions

A square root function involves the square root of an expression. The basic form is \( f(x) = \sqrt{x} \). In these functions, the expression inside the square root, called the radicand, must be non-negative for the function to be real and defined. This means that the values of \(x\) should ensure that the radicand is positive or zero.An example is \( f(x) = \sqrt{4-3x} \). Here, the radicand is \(4-3x\). For this function to have real values, we need \(4-3x \geq 0\). This condition lets us find which \(x\) values are allowed, ensuring the function doesn't produce imaginary numbers. Therefore, handling these conditions correctly is crucial to determine the domain of square root functions.

Inequalities

In mathematics, inequalities express the relative size or order of two values. They are essential in finding domains for functions like square root functions. Inequalities use symbols like \(\geq\), \(>\), \(\leq\), and \(<\).To solve an inequality:

• Perform operations just like equations: add, subtract, multiply, or divide both sides by the same number.
• Remember, when you multiply or divide by a negative number, you must flip the inequality sign.

Using the function from the example, solving \(4-3x \geq 0\) involves these steps. Subtracting \(4\) gives \(-3x \geq -4\), and dividing by \(-3\) flips the inequality to \(x \leq \frac{4}{3}\). These steps ensure we correctly determine for which \(x\) values the function is defined.

Interval notation

Interval notation provides a way to describe the set of all numbers between two endpoints. It's an efficient method to express domains and solutions to inequalities. In interval notation, brackets are important:

• \([a, b]\) means \(x\) includes \(a\) and \(b\).
• \((a, b)\) means \(x\) is between \(a\) and \(b\) but \(a\) and \(b\) are not included.
• Infinite intervals use \(\infty\) or \(-\infty\) and always have parentheses, since infinite values are not actual numbers.

For example, the domain of \(f(x) = \sqrt{4-3x}\) is \(( -\infty, \frac{4}{3} ]\). Here, \(-\infty\) means the domain extends indefinitely to the left, and the bracket \([\) shows that \(\frac{4}{3}\) is included, meaning \(x\) can be \(\frac{4}{3}\) or less.