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1: Relations and Functions

Last updated

May 13, 2022
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- Preface
- 1.1: Sets of Real Numbers and the Cartesian Coordinate Plane

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1.1: Sets of Real Numbers and the Cartesian Coordinate Plane
A brief refresher on some basic notions is welcome, if not completely necessary, at this stage. To that end, we present a brief summary of 'set theory' and some of the associated vocabulary and notations we use in the text.
1.2: Relations
All of Precalculus can be thought of as studying sets of points in the plane. With the Cartesian Plane now fresh in our memory we can discuss those sets in more detail
1.3: Introduction to Functions
One of the core concepts in College Algebra is the function. There are many ways to describe a function and we begin by defining a function as a special kind of relation.
1.4: Function Notation
If we think of the domain of a function as a set of inputs and the range as a set of outputs, we can think of a function f as a process by which each input x is matched with only one output y. Since the output is completely determined by the input x and the process f, we symbolize the output with function notation `f(x) ' . In other words, f(x) is the output which results by applying the process ff to the input x .
1.5: Function Arithmetic
It would seem natural, then, that functions should have their own arithmetic which is consistent with the arithmetic of real numbers. The following definitions allow us to add, subtract, multiply and divide functions using the arithmetic we already know for real numbers.
1.7: Transformations
In this section, we study how the graphs of functions change, or transform, when certain specialized modifications are made to their formulas. The transformations we will study fall into three broad categories: shifts, reflections and scalings, and we will present them in that order.
1.6: Graphs of Functions
In Section 1.3 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most of our time in that section looking at functions graphically because they were, after all, just sets of points in the plane. Then in Section 1.4 we described a function as a process and defined the notation necessary to work with functions algebraically. So now it’s time to look at functions graphically again, only this time we’ll do so with the

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