91Ó°ÊÓ

Suppose \(m\) is an odd integer. Show that the function \(f\) defined by \(f(x)=x^{m}\) is an odd function.

Suppose \(m\) is an even integer. Show that the function \(f\) defined by \(f(x)=x^{m}\) is an even function.

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 \(\left.y=\frac{1}{x} .\right]\)

What is the domain of the function \((3+x)^{1 / 4} ?\)

Suppose \(x>0 .\) Show that the distance from \(\left(x, \frac{1}{x}\right)\) to the point \((\sqrt{2}, \sqrt{2})\) is \(x+\frac{1}{x}-\sqrt{2}\)

What is the domain of the function \(\left(1+x^{2}\right)^{1 / 8} ?\)

Suppose \(x>0\). Show that the distance from \(\left(x, \frac{1}{x}\right)\) to \((-\sqrt{2},-\sqrt{2})\) minus the distance from \(\left(x, \frac{1}{x}\right)\) to \((\sqrt{2}, \sqrt{2})\) equals \(2 \sqrt{2}\)

Suppose \(p\) and \(q\) are rational numbers. Define functions \(f\) and \(g\) by \(f(x)=x^{p}\) and \(g(x)=x^{q} .\) Explain why $$ (f \circ g)(x)=x^{p q} $$

Suppose \(x\) is a real number and \(m, n,\) and \(p\) are positive integers. Explain why $$ x^{m+n+p}=x^{m} x^{n} x^{p} $$

Show that the right triangle with sides of length \(3,4,\) and 5 is the only right triangle in which the side lengths differ by 1 (meaning that the sides have length \(d\), \(d+1\), and \(d+2\) for some number \(d\) ).