91Ó°ÊÓ

Christopher said that \(\mathrm{f}(x)=|x-2|\) and \(\mathrm{g}(x)=|x+2|\) are inverse functions after he showed that \(\mathrm{f}(\mathrm{g}(2))=2, \mathrm{f}(\mathrm{g}(5))=5,\) and \(\mathrm{f}(\mathrm{g}(7))=7 .\) Do you agree that \(\mathrm{f}\) and \(\mathrm{g}\) are inverse functions? Explain why or why not.

Explain the difference between fg \((x)\) and \(f(g(x))\)

Eric said that if \(\mathrm{f}(x)=|2-x|,\) then \(y=2-x\) when \(x \leq 2\) and \(y=x-2\) when \(x > 2 . \mathrm{Do}\) you agree with Eric? Explain why or why not.

Explain why \((x-h)^{2}+(y-k)^{2}=-4\) is not the equation of a circle.

The graph of \(y=x^{2}-4 x+4\) is tangent to the \(x\) -axis at \(x=2\) and does not intersect the \(x\) -axis at any other point. How many roots does this function have? Explain your answer.

Explain the difference between direct variation and inverse variation.

Give an example of a function g for which \(2 \mathrm{g}(x) \neq \mathrm{g}(2 x) .\) Give an example of a function \(\mathrm{f}\) for which \(2 \mathrm{f}(x)=\mathrm{f}(2 x)\)

Can \(y=\sqrt{x}\) define a function from the set of positive integers to the set of positive integers? Explain why or why not.

Kyle said that if \(r\) is directly proportional to \(s,\) then there is some non- zero constant, \(c,\) such that \(r=c s\) and that \(\\{(s, r)\\}\) is a one-to-one function. Do you agree with Kyle? Explain why or why not.

Let \(f(x)=x^{2}\) and \(g(x+2)=x^{2}+2 .\) Are \(f\) and \(g\) the same function? Explain why or why not.