Find the Domain of (10x+2)/√(x²-4): Rational Function Analysis

The given function is $\frac{10x+2}{\sqrt{x^2-4}}$. Finding its domain requires ensuring the denominator is not zero and the expression under the square root is positive.

First, identify the points where the expression under the square root is zero: set $x^2 - 4 = 0$.

Solving $x^2 - 4 = 0$:

• Add 4 to both sides: $x^2 = 4$
• Take the square root of both sides: $x = \pm 2$

This means the points $x = 2$ and $x = -2$ need further inspection since they make the expression zero (hence the denominator would be undefined).

Next, determine where $x^2 - 4 > 0$. This inequality can be rewritten as:

$(x - 2)(x + 2) > 0$

Evaluate the intervals determined by these critical points:

• Interval $x < -2$: Choose $x = -3$, then $(-3-2)(-3+2) = (-5)(-1) = 5 > 0$
• Interval $-2 < x < 2$: Choose $x = 0$, then $(0-2)(0+2) = (-2)(2) = -4 < 0$
• Interval $x > 2$: Choose $x = 3$, then $(3-2)(3+2) = (1)(5) = 5 > 0$

Therefore, the expression is positive in the intervals $x < -2$ and $x > 2$.

To avoid the denominator being zero, these points are not included in the domain, confirming the domain as $x > 2$ and $x < -2$.

Thus, the solution to the problem is $x > 2, x < -2$, which corresponds to choice .