91Ó°ÊÓ

Find a formula for the function \(f\) that assigns to each nonnegative \(x\) the number obtained by dividing \(x\) by 5 , then taking the cube root of the quotient, and finally multiplying the result by the product of \(\frac{1}{2}\) and \(x^{2}\).

Use the zoom feature of a graphics calculator to approximate the coordinates of the points of intersection of \(f\) and \(g\). Zoom until successive values of the \(x\) coordinate have identical first three digits. $$ f(x)=x^{4}-1, g(x)=x^{3}+1 $$

A single respiratory cycle includes one inhalation and one exhalation. During one respiratory cycle of a certain person at rest, the rate of flow \(R\) (in liters per second) of air into a person's lungs at time \(t\) (in seconds) is given by $$ R=0.5 \sin \frac{2 \pi}{5} t $$ a. How long does it take to complete one respiratory cycle? b. How many respiratory cycles are completed in one minute? c. Graph one complete cycle, starting at time \(t=0\). d. Interpret the meaning of positive and negative values of \(R\) e. To the nearest hundredth, find \(R\) when \(t=3\) seconds.

Solve the equation. $$ |x-1|=2 $$

Let \(f(x)=1 / x\). Show that \(f(f(x))=x\) for all \(x \neq 0\).

Can a horizontal line pass through more than one point on the graph of a function? Explain.

Solve the equation. $$ \left|2 x-\frac{1}{2}\right|=\frac{1}{2} $$

The strongest earthquakes ever recorded occurred oft the coast of Ecuador and Colombia in 1906, and in Japan in \(1933 .\) Each had a magnitude of \(8.9 .\) Find the ratio of the amplitude of the largest wave of such a quake to the corresponding amplitude of a zero-level quake.

Find a two-point equation of the given line. The line containing \((3,4)\) and \((1,3)\)

The volume \(V\) of a rectangular box with square base is 60 cubic centimeters. Express the length \(l\) (in centimeters) of a vertical side as a function of the length \(s\) of a side of the base.