Question

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=\frac{x+2}{x}\)

Short answer

The graph of the function \(y=\frac{x+2}{x}\) has no relative extrema and no points of inflection. It is a hyperbola with a hole at x=0. The function approaches 1 as \(x\to±\infty\).

Step-by-step solution

Understanding the Function

First and foremost, it is important to notice that the function \(y=\frac{x+2}{x}\) has a domain that excludes zero (x ≠ 0). As x approaches infinity or negative infinity, y will approach 1. Next, identify the y-intercept by setting x to 0.

Identify Relative Extrema

To find relative extrema, determine the derivative of the function. The derivative of the function is \(y'=\frac{-2}{x^2}\). Setting the derivative equal to zero and solving for x gives no valid solution, hence there are no extrema.

Identify Points of Inflection

Points of inflection can be determined by finding the second derivative and setting it to zero. The second derivative of the function is \(y''=\frac{4}{x^3}\), which never equals to zero. Thus, there are no inflection points.

Sketch the Graph

Finally, sketching the graph involves factoring in all the information gathered: the domain of the function, the behavior as x approaches infinity and negative infinity, the absence of extrema, the absence of inflection points, and the fact that the function approaches 1 as \(x\to±\infty\). It's clear the graph will be a hyperbola with a hole at x=0.