91Ó°ÊÓ

In mathematics, a square root function includes the square root of a variable or an expression. Its general form is \( f(x) = \sqrt{x} \). This function is special because the square root of any number needs to result in a non-negative value. This means that the expression under the square root, called the radicand, must be zero or a positive number. For example, in the function \( f(x) = 3 - \sqrt{6 - 2x} \), the radicand is \( 6 - 2x \). Because you cannot take the square root of a negative value and get a real number, the radicand must be \( 6 - 2x \geq 0 \) for the function to be well-defined. The characteristics of square root functions are relevant when determining domains, as this function imposes natural restrictions on the values that \( x \) can take. This leads us to solving inequalities to find out which values make the radicand non-negative.

When you solve inequalities for a function like \( f(x) = 3 - \sqrt{6 - 2x} \), your primary goal is to identify values of \( x \) that satisfy the expression under the square root being greater than or equal to zero. The inequality for this function is \( 6 - 2x \geq 0 \). The process involves simple algebraic manipulations:

• First, maintain the inequality structure: \( 6 - 2x \geq 0 \).
• To isolate \( x \), subtract \( 6 \) from both sides: \( -2x \geq -6 \).
• Finally, divide each side by \(-2\). Remember that dividing by a negative flips the inequality sign: \( x \leq 3 \).

This gives a clear range of values for \( x \) wherein the original expression \( 6 - 2x \) remains non-negative. Prudent algebraic steps ensure you correctly find the range of valid \( x \) values.

Once you have determined the values for \( x \) that keep the function valid, you express these values using a concise mathematical language known as interval notation. Interval notation is a method of writing subsets of the real number line. In our inequality \( x \leq 3 \), all numbers less than or equal to 3 are part of the domain. Therefore, in interval notation, this domain is expressed as \((-\infty, 3]\).

• The \((-\infty\) symbol means that there is no lower limit to the smallest value \( x \) can be. It stretches to negative infinity.
• The \([3]\) includes the number 3 itself as part of the possible values of \( x \), illustrating that the boundary is closed at 3.

Interval notation presents the function's domain succinctly, providing a clear picture of which real numbers are allowable inputs for the function.