Question

graph the function ( y=\frac{2x}{x^{2}-16} ) by identifying the domain and any symmetries, finding the derivatives ( y ) and ( y ), finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any. find the domain of the function. the domain is (type your answer in interval notation.)

Explanation:

Step1: Find the values that make the denominator zero

Set \(x^{2}-16 = 0\). Using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), where \(a=x\) and \(b = 4\), we have \((x + 4)(x-4)=0\).
Solving \((x + 4)(x - 4)=0\) gives \(x=-4\) or \(x = 4\).

Step2: Determine the domain

The domain of a rational function \(y=\frac{f(x)}{g(x)}\) is all real numbers except the values of \(x\) that make \(g(x)=0\).
For the function \(y=\frac{2x}{x^{2}-16}\), the domain is \((-\infty,-4)\cup(-4,4)\cup(4,\infty)\)

Answer:

\((-\infty,-4)\cup(-4,4)\cup(4,\infty)\)