Question

question 11 let ( f(x) = \frac{x - 3}{sqrt{x^2 + 5}} ). which of the following is the domain of ( f(x) )? (\bigcirc) nothing in this list is correct (\bigcirc) ( (-infty, -5) cup (-5, 5) cup (5, infty) ) (\bigcirc) ( (-infty, infty) ) (\bigcirc) ( (5, infty) ) (\bigcirc) ( (-5, infty) ) (\bigcirc) ( 5, infty) ) (\bigcirc) ( (-infty, -sqrt{5}) cup (-sqrt{5}, sqrt{5}) cup (sqrt{5}, infty) ) (\bigcirc) ( -5, infty) )

Explanation:

Step1: Analyze the denominator

The function is \( f(x)=\frac{x - 3}{\sqrt{x^{2}+5}} \). For the square root in the denominator, the expression inside the square root (\(x^{2}+5\)) must be greater than 0 (since it's in the denominator, it can't be zero).

Step2: Solve the inequality \(x^{2}+5>0\)

We know that for any real number \(x\), \(x^{2}\geq0\). So \(x^{2}+5\geq5>0\) for all real \(x\). This means there are no restrictions on \(x\) from the denominator. Also, the numerator \(x - 3\) is a polynomial, which is defined for all real numbers. So the domain of \(f(x)\) is all real numbers.

Answer:

\((-\infty,\infty)\)