91Ó°ÊÓ

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

Suppose \(a>b \geq 0\). Find a formula in terms of \(x\) 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 \(\left(-\sqrt{a^{2}-b^{2}}, 0\right)\).

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

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

Find an integer \(m\) such that $$ \left((3+2 \sqrt{5})^{2}-m\right)^{2} $$ is an integer.

Find an integer \(m\) such that $$ \left((5-2 \sqrt{3})^{2}-m\right)^{2} $$ is an integer.

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 \(\left(0,-\sqrt{a^{2}+b^{2}}\right)\)

Suppose \(m\) is a positive integer. Explain why \(10^{m}\), when written out in the usual decimal notation, is the digit 1 followed by \(m 0^{\prime}\) s.

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 \(\left(0, \sqrt{a^{2}+b^{2}}\right)\)