- Topics At a Glance
- Variables
- Variables as Unknown Quantities
- Variable Notations
- Constants
- Expressions and Equations
- Rearranging Expressions
- Commutative Properties
- Associative Properties
- Distributive Properties
- Factoring (Distributive Property in Reverse)
- Combining Like Terms
- Eliminating Parentheses
- Simplifying
**Equations, Functions, and Formulas**- Equations
**Functions**- Independent and Dependent Variables
- Formulas
- Applications to Toolbox
- Evaluating Expressions by Substitution
- Evaluating Formulas by Substitution
- Geometric Formulas
- Four-Sided Shapes
- Three-Sided Shapes
- Circles
- Unit Conversion
- Temperatures
- Weights
- Distances and Speeds
- Money
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Functions are probably one of the most important concepts in mathematics. We may say that a lot, but this time we mean it. They are sort of like machines that take our questions and give us answers.

A function may take many inputs and outputs of a single number. The inputs are called the *independent variables.* The number that comes out of the machine is called the *dependent variable*. The term "dependent" should make sense to you because whatever comes out of the machine *depends* on what we put into it. Say you want to make a banana berry smoothie in your blender; you probably shouldn't put raw tuna and Saltines into it. You need to *cook* the tuna.

Consider the grade that you got in a class. You don't need to tell us what it isâ€”you can keep it to yourself. If it was worse than an A++ , we know you will try harder next time. Whatever grade you receive at the end of the semester *depends* on the individual grades you get on each homework assignment, every quiz and your class participation. Your teacher looks at all of these and determines your final grade. The independent variables are the homework scores, the quiz scores, and the extent of classroom participation. The dependent variable is the final grade you receive in the class, because it depends entirely on the independent variables. If you got A's on everything you did in class and then received an F for the semester, that wouldn't make any sense. That would be like you eating all of your vegetables and then not getting any ice cream. *Ripoff*. Anyway, the function machine in this example is your teacher's grading policy.