1.9 Combining Functions (continued)

Piecewise functions

Summary

Consider the function $p$ defined by the following rule:

What are the values of $p(-4)$, $p(-2)$, $p(0)$, $p(2)$, and $p(4)$?

What point is the vertex of the quadratic part of $p$ that is valid for $x \lt 0$? What point is the vertex of the quadratic part of $p$ that is valid for $x \ge 0$?

For what values of $x$ is $p(x) = 0$? In addition, what is the $y$-intercept of $p$?

Sketch an accurate, labeled graph of $y = p(x)$ on the axes provided at left in the following figure.

For the function $f$ defined by the righthand figure in (d), determine a piecewise-defined formula for $f$ that is expressed in bracket notation similar to the definition of $y = p(x)$ above.

Let $r(t) = 2t - 3$ and $s(t) = 5 - 3t$. Determine a formula for each of the following new functions and simplify your result as much as possible.

-$f(t) = (r+s)(t)$-$g(t) = (\frac{s}{r})(t)$-$h(t) = (r \cdot s)(t)$-$q(t) = (s \circ r)(t)$-$w(t) = r(t-4) + 7$

Consider the functions $s$ and $g$ defined by the graphs in Figure and Figure. Assume that to the left and right of the pictured domains, each function continues behaving according to the trends seen in the figures.

• Determine a piecewise formula for the function $y = s(t)$ that is valid for all real numbers $t$. - Determine a piecewise formula for the function $y = g(x)$ that is valid for all real numbers $x$. - Determine each of the following quantities or explain why they are not defined. -$(s \cdot g)(1)$-$(g-s)(3)$-$(s \circ g)(1.5)$-$(g \circ s)(-4)$

One of the most important principles in the study of changing quantities is found in the relationship between distance, average velocity, and time. For a moving body traveling on a straight-line path at an average rate of $v$ for a period of time $t$, the distance traveled, $d$, is given by

In the Ironman Triathlon, competitors swim $2.4$ miles, bike $112$ miles, and then run a $26.2$ mile marathon. In the following sequence of questions, we build a piecewise function that models a competitor's location in the race at a given time $t$. To start, we have the following known information. - She swims at an average rate of $2.5$ miles per hour throughout the $2.4$ miles in the water. - Her transition from swim to bike takes $3$ minutes ($0.05$ hours), during which time she doesn't travel any additional distance. - She bikes at an average rate of $21$ miles per hour throughout the $112$ miles of biking. - Her transition from bike to run takes just over $2$ minutes ($0.03$ hours), during which time she doesn't travel any additional distance. - She runs at an average rate of $8.5$ miles per hour throughout the marathon. - In the questions that follow, assume for the purposes of the model that the triathlete swims, bikes, and runs at essentially constant rates (given by the average rates stated above). - Determine the time the swimmer exits the water. Report your result in hours. - Likewise, determine the time the athlete gets off her bike, as well as the time she finishes the race. - List 5 key points in the form (time, distance): when exiting the water, when starting the bike, when finishing the bike, when starting the run, and when finishing the run. - What is the triathlete's average velocity over the course of the entire race? Is this velocity the average of her swim velocity, bike velocity, and run velocity? Why or why not? - Determine a piecewise function $s(t)$ whose value at any given time (in hours) is the triathlete's total distance traveled. - Sketch a carefully labeled graph of the triathlete's distance traveled as a function of time on the axes provided. Provide clear scale and note key points on the graph. ADD ALT TEXT TO THIS IMAGE ADD ALT TEXT TO THIS IMAGE - Sketch a possible graph of the triathete's velocity, $V$, as a function of time on the righthand axes. Here, too, label key points and provide clear scale. Write several sentences to explain and justify your graph.