Basic Toolkit Functions
In this text, we will be exploring functions—the shapes of their graphs, their unique features, their equations, and how to solve problems with them. 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 elements to build from. We call these our "toolkit of functions"—a set of basic named functions for which we know the graph, equation, and special features.
For these definitions we will use
x as the input variable and
f(x) as the output variable.
Toolkit Functions
- Linear
- Constant: f(x) = c, where c is a constant (number)
- Identity: f(x) = x
- Absolute value: f(x) = |x|
- Power
- Quadratic: f(x) = x2
- Cubic: f(x) = x3
- Reciprocal: $latex f(x) = \frac {1}{x} $
- Reciprocal squared: $latex f(x) = \frac {1}{x^2} $
- Square root: $latex f(x) = \sqrt[2]x = \sqrt x $
- Cube root: $latex f(x) = \sqrt[3]x $
https://youtu.be/sW9-zBeQpCU
You will see these toolkit functions, combinations of toolkit functions, their graphs and their transformations frequently throughout this book. In order to successfully follow along later in the book, it will be very helpful if you can recognize these toolkit functions and their features quickly by name, equation, graph and basic table values.
Not every important equation can be written as
y =
f(x). An example of this is the equation of a circle. Recall the distance formula for the distance between two points: $latex \text{dist} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $
A circle with radius
r with center at (
h,
k) can be described as all points (
x,
y) a distance of
r from the center, so using the distance formula, $latex r = \sqrt{(x - h)^2 + (y - k)^2} $, giving the equation of a circle.
Equation of a Circle
A circle with radius
r with center (
h,
k) has equation $latex r^2 = (x-h)^2 + (y-k)^2 $
Graphs of the Toolkit Functions
Important Topics of this Section
