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. \(y=\frac{x^{2}+1}{x}\)

Short answer

The graph has a vertical asymptote at \(x = 0\) and no horizontal asymptotes, with no critical points or points of inflection. The curve increases from negative infinity to \(-1\), decreases from \(-1\) to \(0\), jumps at \(x = 0\), then decreases from \(0\) to \(1\) and increases from \(1\) to positive infinity.

Step-by-step solution

Rewrite the function

Rewrite the function as \(y=x-\frac{1}{x}\). This format allows us to easily identify the vertical asymptote \(x=0\).

Find the first derivative

Differentiate the function using the power and quotient rules to get \(y' = 1+\frac{1}{x^{2}}\). Set this equal to zero to find the critical points.

Find the critical points

Solving \(y' = 1+\frac{1}{x^{2}} = 0\), we find that there are no real roots. Thus, the function does not have any critical points.

Find the second derivative

Differentiate the derivative to get the second derivative \(y''= -2/x^{3}\). Set this equal to zero to find the points of inflection.

Find the points of inflection

Solving \(y''= -2/x^{3}=0\), we find that there are no real roots. Thus, the function does not have any points of inflection.

Analyze margins

By analyzing the function as \(x\) approaches infinity and negative infinity, we can determine that there are horizontal asymptotes at \(y=\pm\infty\).

Sketch the graph

With this information, we can now sketch the graph of the function. It should show the curve increasing from negative infinity to \(-1\), decreasing from \(-1\) to \(0\), jumping at \(x=0\) because of the vertical asymptote, and then decreasing from \(0\) to \(1\) before increasing from \(1\) to positive infinity.