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Unformatted text preview: Chapter 0: Functions Summary: This chapter sets out to describe mathematical functions. This idea is central to the rest of the book as virtually all concepts will be framed in the context of a function. The important concepts that are covered here are the do- main and range of functions, creating new functions by combining functions and representing functions parametrically. Along the way, families of functions such as polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions are introduced. OBJECTIVES: After reading and working through this chapter you should be able to do the following: 1. Identify a function and its domain and range (§0 . 1). 2. Combine functions arithmetically or through composition to create new functions (§0 . 2). 3. Modify functions through translation, stretching either the inputs or the out- puts or using an appropriate reflection (§0 . 2). 4. Know the properties of power functions, polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions (§0 . 3 , . 5). 5. Find the inverse of a function if it is possible to do so (§0 . 4). 0.1 Functions PURPOSE: To describe what functions are. In this section, the main idea of a function is developed. Here the main discussion is about the domain and range of a function. In particular, functions are described as having the relationship that for each input to a function there is a unique out- input → unique output put value. A clear distinction is made between the inputs and outputs in a function relationship. Inputs are referred to as the independent variable and outputs are the dependent variable. 1 2 Typically, the domain of a function is found by realizing what inputs (or x-values) domain and range will give valid outputs (or y-values). For example, since the square root √ x does not give a real number value for x < 0 then y = √ x would only have a domain of x ≥ 0. The range of the function is directly dependent upon the domain (hence y is the dependent variable). Whatever values are obtained using the valid inputs (the domain) results in the appropriate range. Things to watch out for when inspecting domains are: 1. Division by zero, 2. Functions that only accept certain inputs (i.e., √ x or ln x ), 3. Any restrictions given explicitly. The vertical line test is used to determine if a relationship between x- and y- vertical line test values is actually a function. The convention is that the inputs are usually given along the horizontal axis ( x-axis) and that the outputs are given along the vertical axis ( y-axis). So the vertical line test will indicate when any given input will give you more than one output which would result in a nonfunction. An example is x 2 + y 2 = 1. When x = 1 / 2 then y = √ 3 / 2 or y = − √ 3 / 2. So this fails the vertical line test and is not a function....
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