Problem 61 - College Algebra

Question

In Exercises \(61-64,\) find the domain of each function. $$ f(x)-\sqrt{2 x^{2}-5 x+2} $$

Step-by-Step Solution

Answer

\([0.5, 2]\)

Step 1: Find the Discriminant

The discriminant of the quadratic equation \(2x^2 - 5x + 2\) is found by the formula \(D=b^2-4ac\), where \(a=2, b=-5, c=2\). So, the discriminant is \((-5)^2 - 4*2*2 = 25 - 16 = 9\)

Step 2: Solve the Quadratic Equation

Solve the inequality \(2x^2 - 5x + 2 >= 0\). It's a quadratic equation so use the quadratic formula \(x = [-b ± sqrt(D)] / (2a)\) where D is the discriminant. Substituting the known values, we find that \(x = [5 ± sqrt(9)] / (2*2)\). Therefore, the roots of this quadratic equation are \(x1 = 2\) and \(x2 = 0.5\)

Step 3: Determine the Domain

Since this is a quadratic equation and we are looking for values where it is greater or equal to zero, the domain will be between the roots, including the roots. The domain of the function \(f(x)\) is thus \(x >= 0.5\) and \(x <= 2\), or in interval notation, \([0.5, 2]\)

Key Concepts

Quadratic Equation

A quadratic equation is simply a polynomial equation of the form \(ax^2 + bx + c = 0\). It's called quadratic because the highest exponent of the variable (usually \(x\)) is 2. Every quadratic equation can have up to two solutions or roots. These roots can be found using techniques like factoring, completing the square, or the quadratic formula.

• Quadratic equations give us a parabola when graphed.
• The direction of the parabola (opening upwards or downwards) depends on the sign of \(a\). If \(a > 0\), it opens upwards.

Understanding the structure of a quadratic equation is key to solving them and analyzing their properties.

Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is found using the formula \(D = b^2 - 4ac\). It tells us about the nature of the roots:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root (sometimes called a double root).
• If \(D < 0\), there are two complex roots.

In the given exercise, the discriminant is 9, which is greater than 0, indicating two distinct real roots. The discriminant helps us assess the behavior and graphical representation of the quadratic, assisting in identifying intervals for inequalities.

Inequality Solving

Solving quadratic inequalities involves determining where the quadratic expression is greater than or equal to zero. After finding the roots using the quadratic formula, we graph these solutions on a number line to find where the function is positive or negative.
For the inequality \(2x^2 - 5x + 2 \geq 0\), we found the roots to be 0.5 and 2. The sign of the quadratic changes at these roots, so we test intervals between the roots to determine where the inequality holds.

• Test points in each interval to see if they satisfy the inequality.
• Include the roots if the inequality uses \(\geq\) or \(\leq\).

This approach helps in identifying the domain in cases where square roots are involved.

Interval Notation

Interval notation is a way to represent a range of values, such as the solutions to an inequality. It's concise and captures whether endpoints are included or not using brackets.

• Use square brackets \([ \text{and} ]\) when the endpoints are included (closed interval).
• Use parentheses \(( \text{and} )\) when the endpoints are not included (open interval).

For the function \(f(x) = \sqrt{2x^2 - 5x + 2}\), the domain where the expression under the square root remains non-negative is \([0.5, 2]\). This concise form communicates the values for which the function is defined without redundantly listing them.