91Ó°ÊÓ

Inequality constraints help determine the valid inputs for a function. In this exercise, we are dealing with the function \( f(x) = \sqrt{c - x^2} \). For the square root to be defined, the inside expression \( (c - x^2) \) must be non-negative. We use an inequality to express this requirement. Essentially, we need \( c - x^2 \geq 0 \). This inequality ensures that all values of \( x \), when substituted, do not result in a complex number. Inequalities allow us to understand the limitations on \( c \) and \( x \), ensuring mathematical validity.

The square root function is fundamental here. Square roots are only real numbers if the expression inside them is non-negative. Therefore, for \( f(x) = \sqrt{c - x^2} \) to exist in the real number system, \( c - x^2 \) must be equal to or greater than zero. This inherent property impacts the solution by setting boundaries on permissible values of \( x \) based on \( c \). This is why the inequality \( c - x^2 \geq 0 \) is crucial. When calculating domains, always respect such natural constraints of square root functions to avoid non-real solutions.

Interval notation provides a clear way to describe continuous sets of numbers. In this exercise, the desired domain is expressed as \([-5,5]\), which includes all numbers from \(-5\) to \(5\) inclusive. The square brackets \([]\) signify that the endpoints are included in the interval. This notation is concise and efficiently communicates the domain within which \( x \) must fall in order for \( f(x) \) to be defined. When solving inequalities for a function's domain, converting your solutions into interval notation is a necessary final step for clarity and correct mathematical reporting.

A quadratic expression is an expression of the form \(ax^2 + bx + c\). In our exercise, the term \(-x^2\) is part of a quadratic expression encountered in the domain constraint. Quadratic expressions can form a parabola when graphed. Understanding this shape helps in visualizing how the domain behaves relative to \( c \). Since \( x^2 \) has a minimum value of 0 at \( x = 0 \), for the inequality \( c - x^2 \geq 0 \) to be true across all \( x \) from \(-5\) to \(5\), \( c \) must at least be 25. This value ensures that the parabola is sufficiently shifted upwards to include the entire interval. Quadratics are key to predicting and solving domain constraints in functions involving components like \(-x^2\).