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The square root function involves finding the non-negative root of a given number or expression. For a square root function, like \(f(x) = \sqrt{6 + x - x^2}\), the expression inside the root must be non-negative. This ensures that the output of the square root is a real number and not an imaginary one.
To determine when \(\sqrt{6 + x - x^2}\) is defined, examine the expression inside the square root: \(6 + x - x^2\). This leads us to set up an inequality, ensuring the expression remains non-negative:

• This is the starting point to find the domain of the function.
• By solving \(6 + x - x^2 \geq 0\), we can identify all acceptable \(x\) values.

Breaking down the inner expression properly allows us to understand the limits and behavior of the square root function. This step is crucial, especially when dealing with more complex polynomial expressions within the root.

A quadratic inequality involves a second-degree polynomial being greater or less than zero. In this context, \(6 + x - x^2 \geq 0\) represents a quadratic inequality essential for determining the domain of our square root function.
Simplifying and rearranging the inequality to a standard quadratic form \(-x^2 + x + 6 \geq 0\), and upon multiplying through by -1, it becomes \(x^2 - x - 6 \leq 0\). The inequality direction changes when multiplying by a negative number.

• This form is now easier to solve using methods such as factoring.
• Solutions to the inequality help in revealing intervals where the function is defined.

Quadratic inequalities can be tricky as they deal with ranges of solutions which correspond to specific intervals rather than distinct points. Thus, understanding how to manipulate and solve them is vital for recognizing valid inputs for the function.

Factoring quadratics is a key technique to solve quadratic inequalities or equations. In our example, we are tasked with factoring \(x^2 - x - 6\).
The standard approach is to find two numbers whose product is the constant term (-6) and whose sum is the linear coefficient (-1). For \(x^2 - x - 6\), the factors are \((x - 3)(x + 2) = 0\).

• This forms the foundation to identify "critical points" \(x = 3\) and \(x = -2\), where the polynomial equates to zero.
• These points divide the number line into sections, each of which must be tested in the inequality.

Factoring allows us to split the quadratic into simpler linear factors, making it easy to analyze intervals and solve for the values satisfying the inequality conditions. Without factoring, exploring the quadratic's behavior over different ranges would be immensely difficult.