The Calculus Lifesaver: All the Tools You Need to Excel at Calculus

Book Preface

Trying to do calculus without using functions would be one of the most pointless things you could do. If calculus had an ingredients list, functions would be first on it, and by some margin too. So, the rst two chapters of this book are designed to jog your memory about the main features of functions. This chapter contains a review of the following topics:

– functions: their domain, codomain, and range, and the vertical line test;
– inverse functions and the horizontal line test;
– composition of functions;
– odd and even functions;
– graphs of linear functions and polynomials in general, as well as a brief survey of graphs of rational functions, exponentials, and logarithms; and
– how to deal with absolute values. Trigonometric functions, or trig functions for short, are dealt with in the next chapter. So, let’s kick of with a review of what a function actually is.

Functions

A function is a rule for transforming an object into another object. The object you start with is called the input, and comes from some set called the domain. What you get back is called the output; it comes from some set called the codomain.

Here are some examples of functions:
– Suppose you write f(x) = x2. You have just dened a function f which transforms any number into its square. Since you didn’t say what the domain or codomain are, it’s assumed that they are both R, the set of all real numbers. So you can square any real number, and get a real number back. For example, f transforms 2 into 4; it transforms ?1=2 into 1=4; and it transforms 1 into 1. This last one isn’t much of a change at all, but that’s no problem: the transformed object doesn’t have to be dierent from the original one. When you write f(2) = 4, what you really mean Functions, Graphs, and Lines is that f transforms 2 into 4. By the way, f is the transformation rule, while f(x) is the result of applying the transformation rule to the variable x. So it’s technically not correct to say f(x) is a function”; it should be f is a function.”
– Now, let g(x) = x2 with domain consisting only of numbers greater than or equal to 0. (Such numbers are called nonnegative.) This seems like the same function as f, but it’s not: the domains are dierent. For example, f(?1=2) = 1=4, but g(?1=2) isn’t denned. The function g just chokes on anything not in the domain, refusing even to touch it. Since g and f have the same rule, but the domain of g is smaller than the domain of f, we say that g is formed by restricting the domain of f.
– Still letting f(x) = x2, what do you make of f(horse)? Obviously this is undened, since you can’t square a horse. On the other hand, let’s set h(x) = number of legs x has; where the domain of h is the set of all animals. So h(horse) = 4, while h(ant) = 6 and h(salmon) = 0. The codomain could be the set of all nonnegative integers, since animals don’t have negative or fractional numbers of legs. By the way, what is h(2)? This isn’t dened, of course, since 2 isn’t in the domain. How many legs does a 2″ have, after all? The question doesn’t really make sense. You might also think that h(chair) = 4, since most chairs have four legs, but that doesn’t work either, since a chair isn’t an animal, and so chair” is not in the domain of h. That is, h(chair) is undened.
– Suppose you have a dog called Junkster. Unfortunately, poor Junkster has indigestion. He eats something, then chews on it for a while and tries to digest it, fails, and hurls. Junkster has transformed the food into . . . something else altogether. We could let j(x) = color of barf when Junkster eats x; where the domain of j is the set of foods that Junkster will eat. The codomain is the set of all colors. For this to work, we have to be condent that whenever Junkster eats a taco, his barf is always the same color (say, red). If it’s sometimes red and sometimes green, that’s no good: a function must assign a unique output for each valid input.