Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 9. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | A Pocketful of Functions Some Basic Examples 61 10. Which of the following functions is continuous at x 2 a f x x 3 b f x x_3 c f x x2 __6 11. Which of the following functions is continuous at x 1 a f x 5 b f x 5x c f x . d f x x 1 A POCKETFUL OF FUNCTIONS SOME BASIC EXAMPLES Throughout this text we will be looking at functions analytically and graphically. Consider any equation relating two variables. Every point that lies on the graph of the equation satisfies the equation. Conversely every point whose coordinates satisfy the equation lies on the graph of the equation. The correspondence between pairs of x and y that constitute the coordinates of a point on a graph and that satisfy an equation was a key insight of Rene Descartes 1596-1650 that unified geometry and As a consequence of this insight the fields of geometry and algebra became irrevocably intertwined opening the door to much of modern mathematics including calculus. We have been discussing functions and their graphs in a very general way but it is always strategically wise to include a few simple concrete examples for reference. In this section you will become familiar with a small sampling of functions functions that you can carry about and pull out of your back pocket at any moment. As we go along in the text this collection will grow you will learn that these functions belong to larger families of functions sharing some common characteristics and you ll also be introduced to a greater variety of families of functions. But for the time being we will become familiar with a few individual functions. A Few Basic Examples Consider the following function f x x g x x2 h x x I y x 1 The function f is the identity function its output is identical to its input. The function g is the squaring function its output is obtained by squaring its input. The function h is the absolute value function its output is the magnitude size of its input. In other words if the input is positive or nonnegative
