Problem 49 - Algebra and Trigonometry

Question

Find the domain of the function. $$ h(x)=\sqrt{2 x-5} $$

Step-by-Step Solution

Answer

The domain is \( \left[ \frac{5}{2}, \infty \right) \).

Step 1: Understand the function

The function given is a square root function: \( h(x) = \sqrt{2x - 5} \). A square root function is defined for values where the expression inside the square root sign is non-negative.

Step 2: Set the inside of the square root non-negative

To find where the function is defined, set the inside of the square root (\(2x-5\)) greater than or equal to zero: \[ 2x - 5 \geq 0 \]

Step 3: Solve the inequality

Solve the inequality \( 2x - 5 \geq 0 \) for \( x \). Start by adding 5 to both sides:\[ 2x \geq 5\] Next, divide both sides by 2 to solve for \( x \):\[ x \geq \frac{5}{2} \]

Step 4: Write the domain

The domain includes all values of \( x \) that make the expression \( 2x - 5 \) non-negative, which means \( x \) can be any real number greater than or equal to \( \frac{5}{2} \). So, the domain in interval notation is: \[ \left[ \frac{5}{2}, \infty \right) \]

Key Concepts

Square Root Function

A square root function is an important type of function in algebra that features a square root symbol, typically represented as \( \sqrt{x} \). It describes the set of values that, when squared, yield a given number. For the function \( h(x) = \sqrt{2x - 5} \), this means only values that make \( 2x - 5 \) zero or positive can be used, as square roots of negative numbers are not real. This is why we emphasize the importance of determining non-negative values inside the square root. This is crucial for defining a function's domain because you need real solutions for real world applications.

Inequalities

In the context of determining the domain of a square root function, inequalities help us find which values of \( x \) make the expression under the square root valid. Consider \( 2x - 5 \geq 0 \); solving this inequality ensures that the function \( h(x) = \sqrt{2x - 5} \) returns real numbers.

To solve the inequality, follow these steps:

• First, add 5 to both sides, turning it into \( 2x \geq 5 \).
• Divide each side by 2, resulting in \( x \geq \frac{5}{2} \).

This solution gives us the acceptable values of \( x \) that make the function graphically viable and mathematically correct.

Real Number System

The real number system provides the framework for the quantities we encounter every day. It includes whole numbers, fractions, and decimals – essentially any number that can be found on the number line.

In terms of function domain, real numbers determine the possible inputs that yield real outputs. When working with functions like \( h(x) = \sqrt{2x - 5} \), the restriction \( x \geq \frac{5}{2} \) limits \( x \) to specific real numbers. These are numbers greater than or equal to \( \frac{5}{2} \), meaning the function produces only real results, adhering to the rules of square roots and providing practical applications in the real world.