Helpful Tools for Calculus, Chapter 1: Functions and Relations
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Elementary Functions
At this point we have explored the building blocks of all the main functions. Once we start combining functions we are able to study and interact with more interesting shapes and equations. Combining different functions together creates what are called Elementary Functions. A simple definition of an elementary function is that it is a function built from finite combination of basic functions and operations.
For example, f(x)=sinx+2x-3 is an elementary function because it contains trigonometry and polynomials. We will treat these as is probably obvious, handle each part as you would normally. Domain will be restricted based on whether it works for each part of the function at the same time or not.
A Composite Function is a type of elementary function in which one function takes another function as an input. That is to say, a function is plugged into another function. Piecewise functions are functions that take on different forms depending on the input.
Piecewise Functions
A function that varies its output based on the input could be called a piecewise function. These are generally written using a large curly brace and a row for each section of the domain. For example:
f(x) is a piecewise function with three sections. The right column describes the domain for that section and the left column is the function that occurs on that domain. There shouldn’t be any overlap in domains, otherwise it is not a function. It is proper notation to list each section in order, numerically by domain. Here is an example of a piecewise function written improperly:
This is not acceptable because, although each section is written just fine, the group of them together has overlaps in domain. For example, if x=2.5 then g(x) could be evaluated as 2+x or 5x.
When graphing a piecewise function, we just graph each section one at a time, being such to put dots on the ends of the sections. A filled in dot is used when an x-value is included in the domain, like if 2≤x, and an empty dot is used when the x-value is not included, like 2<x. Below is the example of a graph and its function:
Example 1: Graph the following piecewise function:
Solution:
Composite Functions
A composite function is simply a function with another function inside of it. Take the example of √(x2+3). This is technically a composite function, and we can break it up into its two components:
f(x)= √x
g(x)= x2+3
Thus, we can write the composite function:
Sometimes, writing the interior functions using the function shorthand, with just the letter and not the (x) part, is common to shorten the name of the equation:
Another common notation for composite functions is the symbol ∘. Do not confuse this with the symbol for degrees, °. For clarification, here they are side by side:
Composite ∘ ° Degrees
That notation is used to represent f(g(x))=(f∘g)(x).
Be careful, the order of composition is important and (f∘g)(x)≠(g∘f)(x) generally.
Example 2: 
Applications
Review
An Elementary Function is a function that contains multiple different parent functions
A Piecewise Function is a function where, depending on the input, there are different functions that determine the output
A Composite Function is a function created when a function is plugged into another function