Graphs of simple functions

A function is a process or relationship that connects each element ‘a’ of a non-empty set A to at least one element ‘b’ of another non-empty set B. In Mathematics, a function is a relation f from one set A (the domain of the function) to another set B (the co-domain of the function).

Functions are the foundation of calculus in Mathematics. The functions are the various forms of relationships. In Mathematics, a function is represented as a rule that produces a unique result for each input x. Graphs of simple functions are divided into seven categories, where each is used for a different purpose for the plotting on the graph.

A function’s domain and range are its components.

The domain of a function is the set of all its input values, whereas the range is the function’s possible output.

If there is a function f: A→B such that every element of A is mapped to elements of B, then A is the domain, and B is the co-domain.

Examples of a function:

• x2 (squaring) is a function
• x3+1 is also a function
• Sine, Cosine, and Tangent are functions used in trigonometry
• A sphere’s volume V is proportional to its radius. V = 4/3 r3 express V’s dependence on r.
• A circle’s area can be represented in its radius, A = r2. The radius r affects area A. We say that A is a function of r in the mathematical language of functions.

• The power P wasted in a fixed resistance resistor R is proportional to the current I flowing through the resistor: P= I2R
• The acceleration of a body with a set mass m depends on the force F applied to the body: a = F/m.

Simple functions

A simple function is a finite sum, where the function is a characteristic function on a set A.

A simple function can also be defined as having a finite number of values in its range.

Addition and multiplication bring the collection of simple functions to a close. Furthermore, it is simple to incorporate a simple function.

Step Function vs Simple Function

Simple functions are comparable to step functions. However, they are broader. Step functions are defined on function intervals, which are measurable even when reduced to points. As a result, every step function is a simple function.

Not every simple function is a step function. Simple functions do not have to be defined on an interval; the Dirichlet function is an example of such a function.

Step functions do not exist in some spaces, although they can be replaced with simple functions for analysis.

Types (Forms) of Simple Functions with Graphs:

• Identity Function

When a function returns the same value as the output used as its input, it is an identity function.

An identity function is a function that gives the image of each element in a set B as the same element.

As a result, it has the form g(x) = x and is represented by “I.” Because the image of an element in the domain is identical to the output in the range, it is called an identity function.

As a result, an identity function translates every real number to itself. Because the preimage and image are similar, identity functions are easily determined.

Consider an example of a function that maps elements of set A = {6, 7, 8, 9, 10} to itself. g: A → A such that, g = {(6, 6), (7, 7), (8, 8), (9, 9), (10, 10)}

• Constant function

A constant function is used to express a number that remains constant throughout time, and it is the most straightforward sort of real-valued function.

Constant functions are linear functions with horizontal lines in the plane as their graphs.

Even with different input values, a constant function produces the same output.

What Is the Definition of a Constant Function?

A constant function has the same domain range for all domain values. A straight line parallel to the x-axis is a constant function.

The function’s domain is the x-value displayed on the x-axis, and its range is y or f(x), which is marked concerning the y-axis.

Example of a constant function:

Any function can be termed a constant function if it has the form y = k, where k is a constant and k might be any real number. It is also written as f(x) = k.

It is essential to notice here that the value of f(x) is always ‘k’ and is independent of the value of x. In general, a constant function has the same constant value regardless of the input value.

Here are some instances of constant functions:

• f(x) =1
• F (x) =99
• f(x)= any real number imaginable

Conclusion

A simple function has a finite number of real or complex values inside its range (excluding infinity).

The “measurable” restriction is often eliminated, defining simple functions as those with few possible values. Integration, as well as theories and proofs, are made much easier by these simple functions.

Identity Function and Constant Function inherently express the meaning of simple functions using function graphs.