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A square root function is a type of radical function where the variable is inside a square root sign. In general, it takes the form \(f(x) = \sqrt{g(x)}\), where \(g(x)\) is an expression involving the variable \(x\). Understanding square root functions is crucial because they involve operations that only take real values for non-negative inputs. This means the expression inside the square root must be greater than or equal to zero, ensuring the result is a real number.
The square root function \(\sqrt{x-1}\) is defined only when \(x-1\) is non-negative. This will influence the domain of the function, as square roots of negative numbers are not considered real in standard mathematics.

Inequality solving involves finding the set of values that satisfy an inequality condition. When dealing with a square root function like \(\sqrt{x-1}\), you need to solve an inequality to determine the domain of the function.
To ensure \(x-1\) inside the square root is non-negative, set up the condition \(x-1 \geq 0\). Solve for \(x\) by applying algebraic operations:

• Add 1 to both sides of the inequality: \(x-1 + 1 \geq 0 + 1\)
• This simplifies to: \(x \geq 1\)

This means any \(x\) equal to or greater than 1 makes the expression inside the square root non-negative, allowing us to compute \(\sqrt{x-1}\) accurately.

Interval notation is a concise way to express the set of values that make an inequality true. Once we have determined from the inequality that \(x \geq 1\), we can express this set using interval notation.
Interval notation uses brackets and parenthesis to depict the continuum of numbers:

• The square bracket \([\) indicates that the endpoint is included in the set, known as inclusive.
• The parenthesis \()\) indicates the endpoint is not included, known as exclusive.

In our example, since \(x\) starts at 1 and goes to infinity, this is notated as \([1, \infty)\). Here, "1" is included, as 1 is a valid solution from our inequality, and "\(\infty\)" represents continuous values beyond 1 without an upper bound, not inclusive as infinity itself is not a number.