91Ó°ÊÓ

Find a formula for the described function and state its domain. A rectangle has area \(16 \mathrm{m}^{2}\). Express the perimeter of the rect- angle as a function of the length of one of its sides.

Find a formula for the described function and state its domain. Express the area of an equilateral triangle as a function of the length of a side.

Graph the function \(f(x)=\sqrt{x^{3}+x^{2}+x+1}\) and explain why it is one-to- one. Then use a computer algebra system to find an explicit expression for \(f^{-1}(x) .\) (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

The Heaviside function \(H\) is defined by $$ H(t)=\left\\{\begin{array}{ll}{0} & {\text { if } t<0} \\ {1} & {\text { if } t \geqslant 0}\end{array}\right. $$ It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage \(V(t)\) in a circuit if the switch is turned on at time \(t=0\) and 120 volts are applied instantaneously to the circuit. Write a formula for \(V(t)\) in terms of \(H(t)\). (c) Sketch the graph of the voltage \(V(t)\) in a circuit if the switch is turned on at time \(t=5\) seconds and 240 volts are applied instantaneously to the circuit. Write a formula for \(V(t)\) in terms of \(H(t)\). (Note that starting at \(t=5\) corresponds to a translation.)

(a) If \(g(x)=x^{6}+x^{4}, x \geqslant 0,\) use a computer algebra system to find an expression for \(g^{-1}(x)\) (b) Use the expression in part (a) to graph \(y=g(x), y=x,\) and \(y=g^{-1}(x)\) on the same screen.

Find a formula for the described function and state its domain. A closed rectangular box with volume \(8 \mathrm{ft}^{3}\) has length twice the width. Express the height of the box as a function of the width.

Find a formula for the described function and state its domain. An open rectangular box with volume \(2 \mathrm{m}^{3}\) has a square base. Express the surface area of the box as a function of the length of a side of the base.

Let \(f\) and \(g\) be linear functions with equations \(f(x)=m_{1} x+b_{1}\) and \(g(x)=m_{2} x+b_{2} .\) Is \(f \circ g\) also a linear function? If so, what is the slope of its graph?

If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after \(t\) hours is \(n=f(t)=100 \cdot 2^{t / 3}\). (a) Find the inverse of this function and explain its meaning. (b) When will the population reach \(50,000 ?\)

If you invest \(x\) dollars at \(4 \%\) interest compounded annually, then the amount \(A(x)\) of the investment after one year is \(A(x)=1.04 x .\) Find \(A \circ A, A \circ A \circ A,\) and \(A \circ A \circ A\) o \(A\). What do these compositions represent? Find a formula for the composition of \(n\) copies of \(A .\)