Question

graph the rational function. give the domain and range, and discuss symmetry. give the equations of any asymptotes.\\( f(x) = \frac{1}{x^2 + 2} \\)\
what is the domain of \\( f(x) \\)?\
(type your answer in interval notation. type exact answers, using radicals as needed.)

Explanation:

Step1: Analyze the denominator

The function is \( f(x)=\frac{1}{x^{2}+2} \). For a rational function, the denominator cannot be zero. So we need to check when \( x^{2}+2 = 0 \).

Step2: Solve for x in denominator

Solve the equation \( x^{2}+2 = 0 \). Subtract 2 from both sides: \( x^{2}=- 2 \). But for real numbers, the square of a real number is always non - negative (\( x^{2}\geq0 \) for all real \( x \)), so \( x^{2}=-2 \) has no real solutions. This means the denominator \( x^{2}+2 \) is never zero for any real number \( x \).

Step3: Determine the domain

Since the denominator is defined (non - zero) for all real numbers \( x \), the domain of the function \( f(x) \) is all real numbers. In interval notation, this is \( (-\infty,\infty) \).

Answer:

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