Library of “Toolkit” Functions
In this section, we will explore different functions and the shapes of their graphs. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable.
We might see these toolkit functions and their graphs throughout this book. It will be very helpful if we can recognize them and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.
| Toolkit Functions | ||
|---|---|---|
| Name | Function | Graph |
| Constant | [latex]f\left(x\right)=c[/latex], where [latex]c[/latex] is a constant |  |
| Identity | [latex]f\left(x\right)=x[/latex] |  |
| Absolute value | [latex]f\left(x\right)=|x|[/latex] |  |
| Quadratic | [latex]f\left(x\right)={x}^{2}[/latex] |  |
| Cubic | [latex]f\left(x\right)={x}^{3}[/latex] |  |
| Reciprocal | [latex]f\left(x\right)=\frac{1}{x}[/latex] |  |
| Reciprocal squared | [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex] |  |
| Square root | [latex]f\left(x\right)=\sqrt{x}[/latex] |  |
| Cube root | [latex]f\left(x\right)=\sqrt[3]{x}[/latex] |  |