What it does
A real function takes a real number x and gives back exactly one real number f(x). Two sets go with every function: its domain, the inputs it accepts, and its range, the outputs it actually gives. This page draws the functions you meet in Class 11 maths (relations and functions) and paints those two sets right on the graph:
- the domain as a green band along the x-axis, with a hollow dot where a point is left out;
- the range as an orange band along the y-axis.
The gallery has the identity, a constant, the square and cube (polynomials), the reciprocal 1/x, the modulus |x|, the signum function, the greatest integer function [x], the square root, an exponential 2ˣ and the logarithm log₂ x. Slide x to read off f(x) at any point.
The Combine f and g mode builds f + g, f − g, f × g and f ÷ g from two simple functions and works out the domain of the result: where both f and g are defined, and, for f ÷ g, also where g(x) is not zero.
Before this, the relation checker shows when a relation is a function at all; afterwards, limit zoom zooms in on the jumps of [x] and signum, and the graphing calculator plots anything you can type.
How to use it
- Gallery. Tap a function name. The graph, its rule, domain and range appear. Move the Slide x slider to see the point (x, f(x)) and its value in the box below the graph. Where the function is not defined (1/x at 0, √x for negatives), it says so.
- Combine f and g. Pick a type for f and for g (a x + b, a x² + b, a √x + b, a/x + b or a|x| + b) and type a and b. Then press f + g, f − g, f × g or f ÷ g. f is drawn dashed grey, g dotted orange, and the result in blue. For f ÷ g, red dashed lines mark where g(x) = 0, the x values the result cannot use.
- Check yourself gives fresh questions on [x], signum, domains and (f + g)(a), with the working.
Reading the pictures
- Steps and jumps. For [x] each step starts with a filled dot (that value is included) and ends with an open dot (that value belongs to the next step). Signum has open dots at (0, −1) and (0, 1) and a filled dot at the origin, because signum(0) = 0.
- Single values in the range. The constant function has range {3}, a single dot on the y-axis; signum has three dots; [x] has a dot at every integer.
- Open ends. 1/x never takes the value 0, so its range band has a hole at 0; 2ˣ is always positive, so its band starts just above 0.
Worked example
Open Combine f and g with the starting values f(x) = x² and g(x) = x − 1, and press f ÷ g. The title reads (f ÷ g)(x) = (x²) / (x − 1). A red dashed line stands at x = 1, and the domain box says R − {1}: both functions are defined for every real x, but g(1) = 0. With the slider at x = 1.5, f(1.5) = 2.25 and g(1.5) = 0.5, so (f ÷ g)(1.5) = 2.25 ÷ 0.5 = 4.5, and the page shows exactly that line. Change g’s type to a √x + b with a = 1, b = 0: now the domain is (0, ∞), because √x needs x ≥ 0 and x = 0 makes g zero.
Tips
- is the greatest integer not more than x, so for negative numbers it moves away from zero: [−2.3] = −3.
- A function is many-one when two inputs share an output (x² at 2 and −2), and one-one when no two do (x³). The gallery is a quick way to see this: a horizontal line hits a one-one graph at most once.
- Domain first, then range: the range of f ÷ g can be awkward to find, but the domain only needs “both defined” and “g not zero”.
FAQ
Why is signum(0) = 0 and not undefined? The rule |x|/x would divide by zero, so the definition simply sets signum(0) = 0 by hand. That is why the graph has a filled dot at the origin.
Is this a function or a relation? Every graph here passes the vertical line test: each x gives at most one y, so they are all functions.
Does it save anything? No. It runs in your browser and nothing is stored or sent. fibub is not part of NCERT; the topics follow the Class 11 syllabus, but the words and pictures are our own.
Where the numbers come from
- The standard real functions, their domains and ranges, and the algebra of real functions are standard school mathematics. Wording, pictures and examples are fibub's own; no textbook text, figures or exercises are used.
- Checked against Python's math.floor and numpy.sign for the greatest integer and signum functions, and numpy.roots for the points left out of the domain of f/g.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.