Trigonometry and precalculus
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Functions and their graphs
Revisiting what a function is and how we can define and visualize one.
Introduction to functions
You've already been using functions in algebra, but just didn't realize it. Now you will. By introducing a little more notation and a few new ideas, you'll hopefully realize that functions are a very, very powerful tool.
This tutorial is an old one that Sal made in the early days of Khan Academy. It is rough on the edges (and in between the edges), but it does go through the basic idea of what a function is and how we can define and evaluate functions.
- Introduction to functions
- Function example problems
- Ex: Constructing a function
- Functions Part 2
- Functions as Graphs
- Understanding function notation
- Functions (Part III)
- Functions (part 4)
- Sum of Functions
- Difference of Functions
- Product of Functions
- Quotient of Functions
- Evaluating expressions with function notation
- Evaluating composite functions
- Domain of a function
- Domain of a function
Domain and range
What values can you and can you not input into a function? What values can the function output? The domain is the set of values that the function is defined for (i.e., the values that you can input into a function). The range is the set of values that the function output can take on.
This tutorial covers the ideas of domain and range through multiple worked examples. These are really important ideas as you study higher mathematics.
Function inverses
Functions associate a set of inputs with a set of outputs (in fancy language, they "map" one set to another). But can we go the other way around? Are there functions that can start with the outputs as inputs and produce the original inputs as outputs? Yes, there are! They are called function inverses!
This tutorial works through a bunch of examples to get you familiar with the world of function inverses.
Analyzing functions
You know a function when you see one, but are curious to start looking deeper at their properties. Some functions seem to be mirror images around the y-axis while others seems to be flipped mirror images while others are neither. How can we shift and reflect them?
This tutorial addresses these questions by covering even and odd functions. It also covers how we can shift and reflect them. Enjoy!
- When a function is positive or negative
- Positive and negative parts of functions
- Recognizing Odd and Even Functions
- Connection between even and odd numbers and functions
- Even and odd functions
- Shifting functions
- Shifting and reflecting functions
- Recognizing features of functions (Example 1)
- Recognizing features of functions 2 (example 2)
- Recognizing features of functions 2 example 3)
- Recognizing features of functions 2
- Interpreting features of functions 2 (Example 1)
- Interpreting features of functions 2 (Example 2)
- Interpreting features of functions 2
- Comparing features of functions 2 example 1)
- Comparing features of functions 2 example 2)
- Comparing features of functions 2 example 3)
- Comparing features of functions 2
Undefined and indeterminate answers
In second grade you may have raised your hand in class and asked what you get when you divide by zero. The answer was probably "it's not defined." In this tutorial we'll explore what that (and "indeterminate") means and why the math world has left this gap in arithmetic. (They could define something divided by 0 as 7 or 9 or 119.57 but have decided not to.)
More mathy functions
In this tutorial, we'll start to use and define functions in more "mathy" or formal ways.