Question

Find the domain of the function. $$ f(x)=\sqrt{x^{2}-3 x+2} $$

Short answer

The domain of the function \(f(x)=\sqrt{x^{2}-3 x+2}\) is \((-∞, 1] \cup [2, ∞)\).

Step-by-step solution

Identify the function

The function given is \(f(x)=\sqrt{x^{2}-3 x+2}\). The domain of this function are the values of x for which the function is defined.

Setup the inequality

Because it's a square root function, the operand, in this case the quadratic function \(x^{2}-3 x+2\), must be non-negative. Thus, it's important to solve the inequality \(x^{2}-3 x+2 \geq 0\).

Factor the quadratic function

The quadratic \(x^{2}-3 x+2\) can be factored as \((x - 1)(x - 2)\). Thus, our inequality becomes \((x - 1)(x - 2) \geq 0\).

Solve the inequality

The solutions of the equation \((x - 1)(x - 2) = 0\) are \(x = 1\) and \(x = 2\). We test the intervals \((-∞, 1)\), \((1, 2)\), and \((2, ∞)\) by choosing a test point in each interval and substituting it into the inequality. It is found that the inequality holds for the intervals \((-∞, 1]\) and \([2, ∞)\).

State the domain of the function

Combining the two intervals, the domain of the function \(f(x)\) in interval notation is \((-∞, 1] \cup [2, ∞)\). This means that the function is defined for all x-values less than or equal to 1 and for all x-values greater than or equal to 2.