Functions are a HUGE part of algebra and can be one of the most intimidating concepts to learn.
Think of functions like a chicken. Chickens consume seeds, corn, water, and other things, then they internally process the ingredients and ultimately produce eggs. We then say that EGGS are a FUNCTION of a chicken and what it eats.
In math, the ingredients are variables and/or constants, the chicken is an expression containing the variables, and the eggs equate to the result of evaluating the expression for some set of ingredients.
A function will ALWAYS provide one and only one result for a particular set of data. If more than one result can be obtained from the same input data then it is called a relation and not a function. This is an important distinction that will be discussed in more detail in later sections.
NOTATION. Functions are often written in several ways depending on context. Usually the variables [katex]x,y[/katex] and [katex]z[/katex] are used for the data, or inputs, while terms like [katex]f(x)[/katex] or [katex]g(x,y)[/katex] are generally used to denote the function itself. Sometimes, as an informal shortcut, only a single variable is used to denote the function.
- Formal Notation examples: (this notation is preferred)
- [katex]f(x)=x^{2}+4[/katex]
- [katex]p(y)=y^{2}+\sqrt{8}[/katex]
- [katex]g(x)=3x+4x^{2}-18[/katex]
- Informal, or shortcut examples:
- [katex]y=x^{2}+4[/katex]
- [katex]z=y^{2}+\sqrt{8}[/katex]
- [katex]t=3x+4x^{2}-18[/katex]
- Formal Notation examples: (this notation is preferred)
Functions are often grouped by what the function does or what types of terms are used in the expression side of the function. For example a trigonometric function would contain a term like [katex]\sin{x}[/katex] or [katex]\cos{x}[/katex]. A quadratic function would contain a variable with a power, or exponent of 2.