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When dealing with functions that include a square root, it's crucial to make sure that the expression inside the root is non-negative. This is because the square root of a negative number is not defined in the set of real numbers. In our function, the expression under the square root is \( x-1 \). To ensure this is non-negative, we set up the inequality:\[x-1 \ge 0\]Solving this inequality helps us identify the range of values for \( x \) that are valid under this square root constraint. By adding 1 to both sides of the inequality, we find:\[x \ge 1\]This gives us the first part of our domain restriction, meaning \( x \) must be greater than or equal to 1. It's essential to solve such inequalities before proceeding to identify the function's complete domain.

In any function, the denominator cannot be zero, as division by zero is undefined. For the function \( f(x) = \frac{\sqrt{x-1}}{(x+2)(x-3)} \), we need to check for values of \( x \) that make the denominator zero. The denominator \((x+2)(x-3)\) equals zero when either \(x+2 = 0\) or \(x-3 = 0\).

• Solving \( x+2=0 \) gives \( x = -2 \).
• Solving \( x-3=0 \) gives \( x = 3 \).

To maintain a valid domain, \( x \) cannot be -2 or 3, because the denominator is zero at these values. It's critical to identify these points to ensure our function remains defined across its domain. By excluding these points from the dataset, the function stays valid for all other values.

Interval notation is a concise way to express a range of values. In expressing a domain, interval notation shows which values a function is defined for without listing every number. After solving the square root and denominator constraints, we need to combine these to form the function's domain in interval notation. Given:- \( x \ge 1 \) from the square root constraint- \( x eq 3 \) from the denominator constraintThe interval notation for our function's domain is\[[1, 3) \cup (3, \infty)\]This notation indicates:

• \([1, 3)\) includes all numbers from 1 up to, but not including, 3.
• \((3, \infty)\) includes all numbers greater than 3.

Notice how the union symbol \(\cup\) shows that parts of the domain can be split, representing sections of numbers that work together.

Inequalities play a fundamental role in understanding the domain of a function. These inequalities arise from constraints such as square roots and denominators, affecting the values \( x \) can take. For square roots:- The inequality \( x-1 \ge 0 \) ensures no negative results inside the square root.For denominators:- The inequalities \( x eq -2, 3 \) ensure the expression remains valid by preventing division by zero.The processed inequalities inform how we represent the domain in interval notation, guiding us to include or exclude specific values or intervals. Thus, mastering inequalities allows for a correct representation of function domains. The interplay of inequalities within a function enables an accurate understanding of where a function is defined and operates smoothly.