Question

graph the function ( y=\frac{2 x}{x^{2}-16} ) by identifying the domain and any symmetries, finding the derivatives ( y^{prime} ) and ( y^{prime prime} ), 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. the domain is ( (-infty,-4) cup(-4,4) cup(4, infty) ) (type your answer in interval notation.) identify any symmetries. choose the correct answer below. a. the function is an odd function that is symmetric about the ( y )-axis. b. the function is an even function that is symmetric about the ( y )-axis. c. the function is an even function that is symmetric about the origin. d. the function is an odd function that is symmetric about the origin. e. the function is neither even nor odd.

Explanation:

To check if a function \(y = f(x)\) is even or odd, we use the definitions:

• A function \(y = f(x)\) is even if \(f(-x)=f(x)\) for all \(x\) in the domain.
• A function \(y = f(x)\) is odd if \(f(-x)=-f(x)\) for all \(x\) in the domain.

Given \(y = f(x)=\frac{2x}{x^{2}-16}\), then \(f(-x)=\frac{2(-x)}{(-x)^{2}-16}=\frac{- 2x}{x^{2}-16}=-f(x)\).

Odd - functions have the property of symmetry about the origin. That is, if \((x,y)\) is on the graph of \(y = f(x)\), then \((-x,-y)\) is also on the graph.

Answer:

D. The function is an odd function that is symmetric about the origin.