Problem 73 - A Graphical Approach to College Algebra

Question

Determine the domain of each function. Do not use a calculator. $$f(x)=\sqrt{5+4 x}$$

Step-by-Step Solution

Answer

The domain of the function is \([-\frac{5}{4}, \infty)\).

Step 1: Understand the Function Type

The function given is \( f(x) = \sqrt{5+4x} \). This is a square root function, which means the expression inside the square root, \( 5+4x \), must be greater than or equal to zero for the function to be defined in the real number system.

Step 2: Write the Inequality

Set up the inequality for the expression inside the square root to be greater than or equal to zero: \( 5 + 4x \geq 0 \). This will help us determine the values of \( x \) for which \( f(x) \) is defined.

Step 3: Solve the Inequality

Solve the inequality \( 5 + 4x \geq 0 \). To do this:- Subtract 5 from both sides: \( 4x \geq -5 \).- Divide both sides by 4 to solve for \( x \): \( x \geq -\frac{5}{4} \).

Step 4: Determine the Domain

The solution to the inequality \( x \geq -\frac{5}{4} \) defines the domain of the function. Therefore, the function \( f(x) = \sqrt{5+4x} \) is defined for all \( x \) such that \( x \geq -\frac{5}{4} \).

Step 5: Express the Domain

Express the domain in interval notation as \( [-\frac{5}{4}, \infty) \), which indicates that \( x \) can take any value from \( -\frac{5}{4} \) to infinity, inclusive of \( -\frac{5}{4} \).

Key Concepts

Square Root Function

A square root function usually looks like this: \( f(x) = \sqrt{expression} \). What makes the square root function special is its domain. By domain, we mean the set of all possible input values \( x \). For any square root function to be defined over the real numbers:

• The expression inside the square root (called the radicand) must be non-negative (i.e., greater than or equal to zero).

This means, we can't have a negative number under the square root sign if we're dealing with real numbers, as we don't have real square roots for negative numbers. If the expression inside were negative, the square root would be undefined in real numbers. Understanding this pivotal concept helps solve many real-life problems involving square root functions.

Inequality Solving

Inequalities are mathematical expressions involving the symbols \( <, >, \leq, \geq \). Solving inequalities shares similarities with solving regular equations, but with an important twist. Let's walk through the steps:

• First, treat it like an equation. For example, in \( 5 + 4x \geq 0 \), focus on isolating \( x \).
• Subtract 5 from both sides, giving \( 4x \geq -5 \).
• Next, divide both sides by 4 to solve for \( x \) and you get \( x \geq -\frac{5}{4} \).

Remember, whenever you multiply or divide by a negative number, the inequality sign flips. This doesn't apply here, but it's crucial to bear in mind as it can easily be overlooked. Practice makes perfect with inequalities, ensuring precision in solving and understanding is vital!

Interval Notation

Interval notation is a method for writing the domain of a function. It's a concise way to present the set of numbers that \( x \) can be in. Let's understand its details:

• Brackets \([]\) indicate that a number is included in the solution (closed interval), and parentheses \( () \) show that a number is not included (open interval).
• For example, the domain \( [-\frac{5}{4}, \infty) \) tells us that \( x \) includes \(-\frac{5}{4}\) but goes up indefinitely.
• The infinity symbol (\( \infty \)) is always accompanied by a parenthesis since infinity isn't a real number and can't really be pinpointed.

This notation is extremely useful for efficiently describing the possible ranges for \( x \) in mathematical problems and ensures clarity and simplicity.