Function Basics: Learn It 6

Identifying Basic Toolkit Functions

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 will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this module. It will be very helpful if we can recognize these toolkit functions 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.

You should memorize the shapes of these graphs along with their names in words and in equation form. Be able to plot a few points of each to the left and right of the y-axis. You’ll be using these function families throughout the rest of the module.
Toolkit Functions
Name Function Graph
Constant [latex]f\left(x\right)=c[/latex], where [latex]c[/latex] is a constant Graph of a constant function.
Identity/Linear [latex]f\left(x\right)=x[/latex] Graph of a straight line.
Absolute value [latex]f\left(x\right)=|x|[/latex] Graph of absolute function.
Quadratic [latex]f\left(x\right)={x}^{2}[/latex] Graph of a parabola.
Cubic [latex]f\left(x\right)={x}^{3}[/latex] Graph of f(x) = x^3.
Reciprocal [latex]f\left(x\right)=\frac{1}{x}[/latex] Graph of f(x)=1/x.
Reciprocal squared [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex] Graph of f(x)=1/x^2.
Square root [latex]f\left(x\right)=\sqrt{x}[/latex] Graph of f(x)=sqrt(x).
Cube root [latex]f\left(x\right)=\sqrt[3]{x}[/latex] Graph of f(x)=x^(1/3).