Domain, Range & Function Graphs
Text Calculus is about realvalued functions.
A realvalued function f assigns a real number f(p) to each point p in the domain of the function.
The domain of a function of one variable is a subset of the
real line { x  x ∈ {R} }.
The most common domains are intervals of the form a {leq} x {leq} b.
The range of a realvalued function f is the collection of all real numbers f(x) where x is in the domain of f.
The simplest example of a function is the constant function
that assigns the real number k to all x in the domain. The range of this function is the set { k } containing one point. The next simplest example is a linear function defined by the formula f(x) = px + k where p is the slope of the linear function and k denotes its yintercept. The range of this function will be all real numbers if p is not equal to 0 and just the value { k } if p = 0.
The graph of a function of one variable is the
collection of points (x,f(x)) in the coordinate plane where x is in the domain of f.
The graph of a linear function of one real variable is a line
in the coordinate plane.
Demos
Domain and Range
 
The function f(x) and its domain are given in the control panel. The graph of f(x) is shown in a separate window. There are two red hotpots that can be moved along the xaxis. These two points are used to specify a green domain, whose image under f(x) is shown in cyan. To view the range of f(x) over the green domain, animate the variable called "ShowRange" in the control panel. This pulls the curve onto the yaxis and, therefore, shows the range of f(x) over the green domain.

Exercises 1. What is the range of the function f(x) = ax^{2}? (The answer will depend on the constant a.)
2. What is the range of the function f(x) = x^{4} + 2x^{2}?
