Problem 28 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$h(x)=\sqrt{5 x-3}$$

Step-by-Step Solution

Answer

Domain: \( \left[ \frac{3}{5}, \infty \right) \)

Step 1: Understanding the Function

The function provided is \( h(x) = \sqrt{5x - 3} \). This function is a square root function. For it to be defined, the expression inside the square root must be non-negative.

Step 2: Setting Up the Condition for the Domain

To ensure the square root is defined, set the expression inside the square root greater than or equal to zero: \[ 5x - 3 \geq 0 \]

Step 3: Solving the Inequality

Solve the inequality \( 5x - 3 \geq 0 \):1. Add 3 to both sides: \[ 5x \geq 3 \]2. Divide both sides by 5: \[ x \geq \frac{3}{5} \]

Step 4: Writing the Domain in Interval Notation

From the inequality deduction, \( x \) must be greater than or equal to \( \frac{3}{5} \). Thus, the domain of the function is \[ \left[ \frac{3}{5}, \infty \right) \].

Key Concepts

Square Root Function

A square root function is any function that involves the square root of an expression containing a variable. In mathematics, the function \( h(x) = \sqrt{5x - 3} \) is an example of a square root function. Here, the square root sign \( \sqrt{ \, } \) implies that the expression inside must be non-negative for the function to have real-valued outputs.

When dealing with square root functions, the main goal is to ensure that what's under the square root (known as the radicand) never becomes negative. Negative numbers under a square root result in imaginary numbers, which aren't included in standard real-valued functions.

To determine when the square root function \( h(x) = \sqrt{5x - 3} \) is defined, we set the radicand \( 5x - 3 \) greater than or equal to zero. This keeps our function in the realm of real numbers.

Inequality Solving

Inequality solving is an essential step to finding the domain of functions, especially with expressions under square roots. First, identify the inequality: for the square root function \( h(x) = \sqrt{5x - 3} \), we derive the inequality from the condition that the radicand \( 5x - 3 \) must be non-negative.

Here's how you solve it:

• Start by setting the inequality \( 5x - 3 \geq 0 \).
• Add 3 to both sides, resulting in \( 5x \geq 3 \).
• Finally, divide both sides by 5 to isolate \( x \), giving \( x \geq \frac{3}{5} \).

This process determines the values of \( x \) for which the function is defined within the realm of real numbers. Inequality solving like this helps us understand what inputs (or domain) a function can accept.

Interval Notation

Interval notation is a helpful way to express sets of numbers, mainly used to describe the domain and range of functions. After solving an inequality, we often need to write the solution in this concise form.

For the square root function \( h(x) = \sqrt{5x - 3} \), which is defined when \( x \geq \frac{3}{5} \), we express the domain using interval notation.

• The bracket \( \left[ \right. \) indicates inclusion, meaning \( x = \frac{3}{5} \) is part of the domain.
• The symbol \( \infty \) represents infinity, used since there is no upper boundary for \( x \).
• Combining these, the domain is \( \left[ \frac{3}{5}, \infty \right) \), indicating that \( x \) starts from \( \frac{3}{5} \) and extends to infinity.

This notation is a compact and clear way to show where the function is defined, or where it "lives" on the number line.