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) \\)?\\( (-\infty, \infty) \\)\
(type your answer in interval notation. type exact answers, using radicals as needed.)\
identify any vertical asymptotes. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\
\\( \bigcirc \\) a. there is one vertical asymptote. its equation is \\( \square \\).\
(type an equation. type an exact answer, using radicals as needed.)\
\\( \bigcirc \\) b. there are two vertical asymptotes. the equation of the leftmost one is \\( \square \\) and the equation of the rightmost one is \\( \square \\).\
(type equations. type exact answers, using radicals as needed.)\
\\( \bigcirc \\) c. there are no vertical asymptotes.

Explanation:

For the domain:

Step1: Analyze the denominator

The function is \( f(x)=\frac{1}{x^2 + 2} \). The denominator is \( x^2+2 \). We need to find when the denominator is zero (since division by zero is undefined) to determine restrictions on the domain.
Solve \( x^2 + 2=0 \). Subtract 2 from both sides: \( x^2=- 2 \). Since the square of a real number cannot be negative, there are no real values of \( x \) that make the denominator zero.

Step2: Determine the domain

Since there are no restrictions on \( x \) (no real \( x \) makes the denominator zero), the domain is all real numbers. In interval notation, that is \( (-\infty,\infty) \).

For the vertical asymptotes:

Step1: Recall the definition of vertical asymptotes

Vertical asymptotes occur where the function is undefined (denominator is zero) and the numerator is not zero (or the function has a non - removable discontinuity).

Step2: Analyze the denominator of \( f(x)=\frac{1}{x^2 + 2} \)

We already saw that \( x^2+2 = 0 \) has no real solutions (because \( x^2=-2 \) has no real roots). So, there are no values of \( x \) for which the function is undefined due to a zero denominator (in the real number system). Therefore, there are no vertical asymptotes.

Answer:

(for domain):
\((-\infty, \infty)\)