91Ó°ÊÓ

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\). $$ f(x)=\frac{2}{x} $$

Suppose \(b>a>0 .\) Find a formula in terms of \(y\) for the distance from a typical point \((x, y)\) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) to the point \(\left(0,-\sqrt{b^{2}-a^{2}}\right)\).

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\). $$ f(x)=\frac{3}{x^{2}} $$

Suppose \(a\) and \(b\) are nonzero numbers. Find a formula in terms of \(y\) for the distance from a typical point \((x, y)\) with \(y>0\) on the hyperbola \(\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1\) to the point \(\left(0,-\sqrt{a^{2}+b^{2}}\right)\).

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\). $$ f(x)=-\frac{2}{x^{2}} $$

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\). $$ f(x)=-\frac{3}{x} $$

Suppose \(a\) and \(b\) are nonzero numbers. Find a formula in terms of \(y\) for the distance from a typical point \((x, y)\) with \(y>0\) on the hyperbola \(\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1\) to the point \(\left(0, \sqrt{a^{2}+b^{2}}\right)\).

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\). $$ f(x)=-\frac{2}{x^{2}}+3 $$

Suppose \(x>0\). Show that the distance from \(\left(x, \frac{1}{x}\right)\) to the point \((-\sqrt{2},-\sqrt{2})\) is \(x+\frac{1}{x}+\sqrt{2}\). [See Example 6 in Section 2.3 for a graph of \(\left.y=\frac{1}{x} .\right]\).

Sketch the graph of the given function \(f\) on the domain \(\left[-3,-\frac{1}{3}\right] \cup\left[\frac{1}{3}, 3\right]\). $$ f(x)=-\frac{3}{x}+4 $$