91Ó°ÊÓ

A quadratic inequality involves a polynomial of degree two, typically expressed in the form \(ax^2 + bx + c \geq 0\) or \(ax^2 + bx + c \leq 0\). To determine where this quadratic function is non-negative or non-positive, you solve it like a normal equation and then test the intervals to see where the inequality holds true.

First, find the roots of the quadratic equation \(4x^2 - 9x + 2 = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions, which are the points where the graph crosses the x-axis.

• Calculate the discriminant \(b^2 - 4ac\). A positive discriminant indicates two real roots.
• Calculate the x-values using the quadratic formula.

Next, examine the intervals defined by these roots. Substitute test points from the intervals into the inequality \(4x^2 - 9x + 2 \geq 0\) to find which intervals satisfy the inequality, hence determining where the expression is non-negative.

This interval testing helps ensure we're only considering x-values where the quadratic expression doesn't make the square root in the function undefined or imaginary.

Radical functions contain a variable within a root, often a square root. Here, we have \(f(x) = \frac{1}{\sqrt{4x^2 - 9x + 2}}\), where the expression inside the square root, \(4x^2 - 9x + 2\), must be non-negative for the function to be defined.

Radical expressions are only defined when the radicand (the expression under the radical) is greater than or equal to zero. This ensures that we're not trying to find the square root of a negative number, which would not yield a real result.

• If the expression is 0, the function becomes undefined due to division by zero.
• If the expression is negative, the result inside the square root would be undefined in the real number system.

Ensuring \(4x^2 - 9x + 2 \geq 0\) is necessary so that we only include x-values that render the expression non-negative, thus allowing the function to exist in real numbers without being undefined.

For a function like \(f(x) = \frac{1}{\sqrt{4x^2 - 9x + 2}}\), undefined points occur where any part of the function leads to division by zero or an invalid operation. In this case, both possibilities arise:

1. **Division by Zero:** Since \(\frac{1}{\text{something}}\) is undefined when the denominator is zero, solving \(\sqrt{4x^2 - 9x + 2} = 0\) reveals x-values that lead to a zero denominator. These x-values must be excluded from the domain.
2. **Negative Inside a Square Root:** The expression \(\sqrt{4x^2 - 9x + 2}\) is undefined for negative numbers as they don't have real square roots, restricting the function's domain further.

• Identify points of zero and solve where the radicand equals zero.
• Exclude these x-values from the set of possible inputs to the function.

By doing both, ensure you're capturing all placeholders where the function cannot process real number values, thus defining a valid domain.