91Ó°ÊÓ

Suppose that both \(f\) and \(g\) have inverses and that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Show that \(h\) has an inverse given by \(h^{-1}=g^{-1} \circ f^{-1}\).

What relationship between \(a, b\), and \(c\) must hold if \(x^{2}+a x+y^{2}+b y+c=0\) is the equation of a circle?

. Let \(f(x)=\frac{3 x+2}{x^{2}+1}\) and \(g(x)=\frac{1}{100} \cos (100 x)\). (a) Use functional composition to form \(h(x)=(f \circ g)(x)\), as well as \(j(x)=(g \circ f)(x)\). $$ \begin{array}{l} \text { (a) Use functional composition to form } h(x)=(f \circ g)(x), \text { as }\\\ \text { well as } j(x)=(g \circ f)(x) \text { . } \end{array} $$

find the best decimal approximation that your calculator allows. Begin by making a mental estimate $$ (3.1415)^{-1 / 2} $$

Find \(\delta\) (depending on \(\varepsilon\) ) so that the given implication is true. $$ |x-2|<\delta \Rightarrow|4 x-8|<\varepsilon $$

Let \(f(x)=\frac{a x+b}{c x+d}\) and assume \(b c-a d \neq 0\). (a) Find the formula for \(f^{-1}(x)\). (b) Why is the condition \(b c-a d \neq 0\) needed? (c) What condition on \(a, b, c\), and \(d\) will make \(f=f^{-1}\) ?

find the best decimal approximation that your calculator allows. Begin by making a mental estimate $$ \sqrt{8.9 \pi^{2}+1}-3 \pi $$

Find \(\delta\) (depending on \(\varepsilon\) ) so that the given implication is true. $$ |x+6|<\delta \Rightarrow|6 x+36|<\varepsilon $$

Suppose that a continuous function is periodic with period 1 and is linear between 0 and \(0.25\) and linear between \(-0.75\) and \(0 .\) In addition, it has the value 1 at 0 and 2 at \(0.25 .\) Sketch the function over the domain \([-1,1]\), and give a piecewise definition of the function.

Suppose that a continuous function is periodic with period 2 and is quadratic between \(-0.25\) and \(0.25\) and linear between \(-1.75\) and \(-0.25 .\) In addition, it has the value 0 at 0 and \(0.0625\) at \(\pm 0.25 .\) Sketch the function over the domain \([-2,2]\), and give \(a\) piecewise definition of the function.