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The domain of a function is the complete set of possible values of the independent variable that make the function work without any issues. For instance, if we look at the function \( f(x) = \sqrt{25 - x^2} \), the inner expression \( 25 - x^2 \) must be non-negative, otherwise, the square root becomes undefined for real number operations.

Hence, not every value of \( x \) will work. To find the domain, we solve for values of \( x \) such that \( 25 - x^2 \geq 0 \). This process involves isolating the variable \( x \) until we have a clear inequality that describes valid \( x \) values. This gives us the range of \( x \) values for which the original function \( f(x) \) is defined, in this case, the interval \([-5, 5]\).

Key points to remember:

• Always check under a square root or in a denominator for problematic values.
• Solve inequalities or equations that arise from these conditions.
• The result provides the domain, which may be one interval or a combination of intervals.

Inequality solving is a method used to find the set of values for a variable that satisfies an inequality. In our example, we need to determine where \( 25 - x^2 \geq 0 \). Inequalities can usually be solved by manipulating them much like equations:

First, ensure the inequality sign is correct for your original statement. Then, rearrange and simplify as needed. In this example, we rearrange to \( 25 \geq x^2 \) (or \( x^2 \leq 25 \)).

Next, take the square root of both sides to find \( -5 \leq x \leq 5 \). These bounds show where the inequality holds true. Intervals obtained from inequalities are critical for determining the eventual domain of the function.

Solutions to inequalities:

• Solve them step-by-step like equations, respecting the inequality signs.
• Create intervals based on where the inequality holds true or is solved.
• Consider these intervals when examining domains or ranges of functions.

A square root function involves an expression that requires a non-negative quantity under the square root symbol. The expression \( f(x) = \sqrt{25 - x^2} \) is only defined when \( 25 - x^2 \) is zero or positive because square roots of negative numbers do not yield real results (excluding complex numbers).

When working with square roots:

• Set up conditions inside the root to ensure they are non-negative.
• Solve these conditions to find valid inputs.
• Understand that the square root function is only defined for non-negative inputs.
• Square root functions often result in a half-parabolic graph limited to one side of the x-axis.

This provides us with an algebraic function curve, but only where the inner expression can genuinely exist to provide real number results.

Critical points in this context are values in the domain where the expression related to a function changes behavior, such as changing sign or hitting zero. In the function \( f(x) = \sqrt{25 - x^2} \), critical points appear wherever the expression \( 25 - x^2 \) equals zero.

For this particular function, these critical points are at \( x = -5 \) and \( x = 5 \). They are crucial because:

• They represent points where the expression inside the square root reaches zero.
• Critical points help set bounds for the domain since outside of these points, the expression could become negative.
• They act as breaks in the intervals on a sign chart, illustrating where \( 25 - x^2 \) changes from positive to negative.

To handle critical points:

• Identify where the expression hits zero.
• Test intervals around these points if necessary.
• Use these points to define the boundaries of valid input ranges.

This method ensures a full understanding of where the function is valid and how it behaves within its domain.