Finding the Domain of a Square Root Function

To find the domain of the function \(f\left( x \right) = \sqrt {\left( {2 - x} \right)\left( {x - 3} \right)} \), we need to ensure that the expression inside the square root is non-negative. The domain of a function involving a square root \(\sqrt{g(x)}\) is defined for all values of \(x\) where \(g(x) \ge 0\).

Setting up the Inequality

In this case, \(g(x) = \left( {2 - x} \right)\left( {x - 3} \right)\). So, we must solve the inequality:

\[ \left( {2 - x} \right)\left( {x - 3} \right) \ge 0 \]This inequality involves a product of two linear factors. To make the analysis easier, we can rewrite the first factor \((2-x)\) as \(-(x-2)\). The inequality then becomes:

\[ -\left( {x - 2} \right)\left( {x - 3} \right) \ge 0 \]

Solving the Inequality

To eliminate the negative sign, we can multiply both sides of the inequality by -1. Remember that multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign.

\[ \left( {x - 2} \right)\left( {x - 3} \right) \le 0 \]

Now we need to find the values of \(x\) for which the product \((x - 2)(x - 3)\) is less than or equal to zero.

Identifying Critical Points

The critical points are the values of \(x\) where the factors are zero. Setting each factor equal to zero:

\(x - 2 = 0 \implies x = 2\)

\(x - 3 = 0 \implies x = 3\)

These critical points, 2 and 3, divide the number line into three intervals: \((-\infty, 2)\), \((2, 3)\), and \((3, \infty)\). We need to check the sign of \((x - 2)(x - 3)\) in each interval.

Analyzing Intervals using a Sign Table

We can create a sign table to analyze the sign of the expression \((x - 2)(x - 3)\) in each interval:

Determining the Domain

We are looking for where \(\left( {2 - x} \right)\left( {x - 3} \right) \ge 0\). From the sign table, this occurs when \(x\) is in the interval \((2, 3)\). We also need to consider the critical points themselves. At \(x=2\) and \(x=3\), the expression \(\left( {2 - x} \right)\left( {x - 3} \right)\) equals 0, which satisfies the condition \(\ge 0\). Therefore, the critical points are included in the domain.

Combining the interval \((2, 3)\) with the critical points 2 and 3, the domain is the closed interval \([2, 3]\).

The domain of the function \(f\left( x \right) = \sqrt {\left( {2 - x} \right)\left( {x - 3} \right)} \) is \([2, 3]\).

Revision Table: Understanding Domain

Additional Information: Solving Quadratic Inequalities

The inequality \((x-2)(x-3) \le 0\) is a type of quadratic inequality. Here are common methods to solve them:

Sign Table Method: As demonstrated above, identify critical points, divide the number line into intervals, and test the sign of the expression in each interval.

Graphing Method: Sketch the graph of the quadratic function \(y = (x-2)(x-3)\). This is a parabola opening upwards with roots at \(x=2\) and \(x=3\). The inequality \((x-2)(x-3) \le 0\) is satisfied where the graph is below or on the x-axis. For an upward-opening parabola, this is between the roots, including the roots, i.e., \([2, 3]\).

Test Point Method: Choose a test point in each interval defined by the critical points and substitute it into the inequality to see if it holds true.

Understanding how to solve inequalities, especially those involving polynomials, is crucial for finding the domain of many types of functions, including square root functions and rational functions.