Problem 10 - College Algebra

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 is \( (-\infty, \frac{4}{3}] \).

Step 1: Understand the Function Type

The function given is \( f(x) = \sqrt{4 - 3x} \). This is a square root function, which means the expression inside the square root, \( 4 - 3x \), must be non-negative.

Step 2: Set Up the Inequality

For the square root to be a real number, the expression inside the root must be greater than or equal to zero: \( 4 - 3x \geq 0 \).

Step 3: Solve the Inequality

Solve \( 4 - 3x \geq 0 \) for \( x \).\[4 - 3x \geq 0 \4 \geq 3x \\frac{4}{3} \geq x \x \leq \frac{4}{3}\]

Step 4: Write the Domain in Interval Notation

The solution to the inequality is \( x \leq \frac{4}{3} \). Therefore, the domain of \( f(x) \) in interval notation is \( (-\infty, \frac{4}{3}] \).

Key Concepts

Square Root Function

The square root function is quite common in mathematical exercises. It is often represented as \( f(x) = \sqrt{g(x)} \), where \( g(x) \) is a function inside the square root. In mathematics, for a square root to be defined and yield real numbers, the expression within the root must be non-negative. This is because square roots of negative numbers are not real in the set of real numbers.
In the given exercise, we have \( f(x) = \sqrt{4 - 3x} \). The function inside the square root is \( 4 - 3x \), which must be greater than or equal to zero for the square root to be defined.
To find out when this occurs, we examine the inequality \( 4 - 3x \geq 0 \). This leads us to determine the domain of the function.

Interval Notation

Interval notation is a common way to present solutions for domains and ranges of functions. It uses brackets to indicate which parts of a function's domain are included or excluded.
In interval notation:

• An open bracket \( ( \) or \( ) \) indicates that the endpoint is not included in the interval.
• A closed bracket \( [ \) or \( ] \) includes the endpoint in the interval.

For instance, in our solution, the domain of the function for \( f(x) = \sqrt{4 - 3x} \) is \( (-\infty, \frac{4}{3}] \). Here:

• The function is defined for all \( x \) from negative infinity up to \( \frac{4}{3} \) (indicated by the closed bracket).

Using interval notation helps in easily expressing a range of values that variables can take considering all restrictions in the function's domain.

Inequality Solutions

Inequalities are expressions of mathematical statements that involve the symbols \( > \), \( < \), \( \geq \), or \( \leq \). Solving inequalities is crucial when finding the domain of functions, especially those involving square roots.
From our exercise, to ensure \( 4 - 3x \geq 0 \), we need to find the values of \( x \) that satisfy this inequality. We solve it step-by-step:

• Start by writing the inequality: \( 4 - 3x \geq 0 \).
• Rearrange the terms to isolate \( x \): \( 4 \geq 3x \).
• Divide each side by 3 to solve for \( x \): \( x \leq \frac{4}{3} \).

This solution tells us that \( x \) can be any number less than or equal to \( \frac{4}{3} \). By solving the inequality, we find where the function is defined and ensure that \( f(x) \) remains a real number across its domain.