Find the domain of the function f(x)=(1)/(√(2 x^2+5 x+2)) (1) R (2) (-2, (-1)/(2)) (3) (-∞,-2] ∪

step 1 Hello students in this problem we have to find the domain of this function So I will apply the c

Find the domain of the function f(x)=(1)/(√(2 x^2+5 x+2))...

Find the domain of the function $f(x)=\frac{1}{\sqrt{2 x^{2}+5 x+2}}$
(1) $\mathrm{R}$
(2) $\left(-2, \frac{-1}{2}\right)$
(3) $(-\infty,-2] \cup\left[\frac{-1}{2}, \infty\right)$
(4) $(-\infty,-2) \cup\left(\frac{-1}{2}, \infty\right)$

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Key Concepts

Domain of a Function

The domain of a function is the set of all input values (usually x-values) for which the function is defined. It is important to identify any restrictions such as division by zero or the need for the radicand of a square root to be non-negative, ensuring that the output of the function is a real number.

Radical Functions and Domain Restrictions

Functions that involve square roots have the additional condition that the radicand (the expression inside the square root) must be non-negative to produce real numbers. Furthermore, if the square root appears in the denominator, the radicand must be strictly positive to avoid division by zero.

Quadratic Inequalities

Quadratic inequalities involve determining the range of values that satisfy an inequality with a quadratic expression. Solving these often requires finding the roots of the quadratic equation, then analyzing the sign of the quadratic expression across the intervals determined by these roots.

Factorization of Quadratic Expressions

Factorizing a quadratic expression simplifies the process of finding its zeros. By writing the quadratic in a factored form, one can easily identify the critical points where the expression is zero and subsequently determine the intervals where the expression is positive or negative, which is key when solving inequalities.

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