Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x²+4)

Verified step by step guidance

1. Identify the denominator of the rational function \(r(x) = \frac{x}{x^2 + 4}\). The denominator is \(x^2 + 4\).
2. Set the denominator equal to zero to find values of \(x\) that might cause vertical asymptotes or holes: solve \(x^2 + 4 = 0\).
3. Solve the equation \(x^2 + 4 = 0\) by isolating \(x^2\): \(x^2 = -4\). Since \(x^2\) cannot be negative for real numbers, there are no real solutions.
4. Since there are no real values of \(x\) that make the denominator zero, the function has no vertical asymptotes.
5. Check for holes by seeing if any factors cancel between numerator and denominator. Here, the numerator is \(x\) and the denominator is \(x^2 + 4\), which share no common factors, so there are no holes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator equals zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. To find them, set the denominator equal to zero and solve for x, excluding any values that also make the numerator zero.

Holes in the Graph

Holes occur in the graph of a rational function when a factor cancels out from both numerator and denominator, resulting in an undefined point rather than an asymptote. Identifying holes involves factoring and simplifying the function to see if any common factors exist.

Rational Functions and Domain Restrictions

Rational functions are ratios of polynomials, and their domain excludes values that make the denominator zero. Understanding domain restrictions helps determine where the function is undefined, which is essential for locating vertical asymptotes and holes.