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Square root functions are a type of radical function specifically involving the square root of a variable. The general form is given by \( f(x) = \sqrt{x} \). However, they can be more complex, like \( g(x) = \sqrt{x - a} \), which represents a horizontal shift of the function. This modification affects where the domain begins.

For the square root to be real, the expression inside must be non-negative, which means that you only use x values that make the radicand (the expression inside the square root) zero or positive. For example, for \( g(x) = \sqrt{x-5} \), the radicand is \( x - 5 \).

To find for which x values the function is defined, you solve the inequality: \( x - 5 \geq 0 \). Solving this gives \( x \geq 5 \), defining the domain of the function. Thus, only inputs from 5 onwards will yield real outputs, forming the complete set of x-values that keep the function within its valid range.

Sketching the graph of a square root function helps visualize the behavior over its domain. It follows a distinctive shape: a curve starting at a particular point and increasing outward. Here’s how to approach it:

• Start with the point where the domain begins. For example, \( x = 5 \) for \( g(x) = \sqrt{x-5} \). At \( x = 5 \), the function \( g(5) = 0 \), starting our curve at point \( (5, 0) \).
• Select a few more x-values to find additional points. When \( x = 6 \), \( g(6) = 1 \); and for \( x = 9 \), \( g(9) = 2 \).

Connect these points with a smooth curve. The graph gradually slopes upwards and away from the origin, resembling half of a parabola.

This function does not exist to the left of \( x = 5 \) and continues upwards with a gradual bend toward higher x-values, indicating it only opens to the right.

Solving inequalities is crucial when determining the domain of functions involving square roots, as it ensures the radicand stays non-negative. Here, solving \( x - a \geq 0 \) clarifies where the function is valid.

To solve \( x - 5 \geq 0 \), rearrange to find \( x \geq 5 \). This tells us the smallest x-value that keeps the expression inside the square root, \( x - 5 \), non-negative.

• Check boundary conditions precisely. The point where \( x = 5 \) makes the expression zero, it's included (hence the \( \geq \) sign).
• Expressing the solution helps define the function’s domain in inequality form, supporting graph plotting and understanding.

Understanding and solving these inequalities is essential for accurately describing where a square root function exists and how it behaves across its domain.