A function in mathematics is a specific type of relation with some rules. A function f, for example, is a relation from a non-empty set A to a non-empty set B where the domain of f is A and no two distinct ordered pairs in f have the same first element. Furthermore, we have various types of functions that can be defined based on their properties.

Functions: A function is a relationship between a set of inputs and a set of outputs, with each input being related to exactly one output.

Assume we have the sets P and Q. Mapping from P to Q will be a function only if each element in the set P has only one element from the set Q assigned to it.

Graphs: In mathematics, and more specifically in graph theory, a graph is a structure consisting of a set of objects, some of which are “related” in some way. The objects correspond to mathematical abstractions known as vertices, and each pair of related vertices is known as an edge. A function’s graph is the set of all points in the plane of the form (x, f(x)). The graph of f could also be defined as the graph of the equation y = f. (x). As a result, the graph of a function is a subset of the graph of an equation.

## Function types and their graphs

There are several types of algebraic functions, each with its own set of characteristics. Many of these algebraic functions can be identified simply by inspecting their graph.

Depending on the type of function graphed, different types of graphs exist.

## Graph of Linear Function

Graph of Squaring Function

Graph of Cubic Function

Graph of Square Root Function

Graph of Reciprocal Function

Graph of Step Function

Graph of Piece-Wise Function

## 1. Graph of Linear Function

All linear functions have the form f(x)=ax+b, where a and b are real numbers and an is nonzero. These functions’ graphs will always be straight lines. This means that the linear function is a first-degree polynomial function:

## 2. Graph of Squaring Function

A squaring function graph is commonly referred to as a parabola, which is a U-shaped curve. The squaring function f(x) = x2. The diagram is shown below.

## Squaring Function Properties

The domain of the squaring function is the set of all real numbers corresponding to the x-axis.

Because the graph is U-shaped, the squaring function’s range is all non-negative real numbers.

Because it is symmetric along the y-axis, the function is an even function.

The squaring function’s intercept is at point (0, 0).

The squaring function graph has a relative minimum at (0, 0).

## 3.Graph of Cubic Function

In contrast to the squaring function, the graph of the cubic function has both positive and negative sides. f(x) = x3

The graph of a cubic function looks like this.

## Cubic Function Properties

The cubic function is a peculiar function.

Along the origin, the cubic function is symmetric.

Cubic function’s domain is the set of all real numbers.

Because the function has an interval between, the range of the cubic function is a set of all real numbers (-∞,∞)

The graph’s intercept is located at (0, 0).

## 4.Graph of Square Root Function

A graph of the function f(x), where x results in x.

f(x) = x

The squaring function graph is shown below.

As we can see, the graph is only on the positive side for both x and f(x).

### Squaring Function Properties

The domain consists of a set of non-negative real numbers (0, ∞).

The range consists of a collection of non-negative real numbers (0, ∞).

The graph’s intercept is located at (0, 0).

It rises between the intervals (0,∞).

## 5. Graph of Reciprocal Function

Let f(x) be a function f(x) = 1x is known as a reciprocal function. The reciprocal graph is shown below.

Although the reciprocal function is symmetric along the origin, it never touches it.

## 6. Graph of Step Function

The step function graph resembles a stairwell with steps.

Let f(x) = [[x]] + 1 Denote the step function. [[x]] denotes a value that is ‘greater than or equal to x’.

The step function graph is shown below.

### Step Function properties

The domain is a collection of all real numbers.

The function’s range is a set of all integers.

The x-intercept is [0, 1] and the y-intercept is (0, 0).

Between each pair of integers, the graph remains constant.

For each y-value, the graph jumps one unit vertically.

## 7. Graph of Piece-Wise Function

Piecewise functions, also known as functions defined by parts, are functions that have different expressions depending on where the independent variable is found in the interval.

When x is greater than or equal to 0, f(x) equals x, and when x is less than 0, f(x) equals -x.The graph of f(x) then has two pieces, one of which corresponds to x (when x is in the interval (0, ∞) and the other to -x (when x is in the interval (-∞, 0).

The Piece-Wise Function is illustrated below.

## Conclusion

In this article we conclude that, Graphing functions is the method of drawing the graph of the corresponding function. Graphing basic functions such as linear, quadratic, cubic, and so on is fairly simple; however, graphing complex functions such as rational, logarithmic, and so on requires some skill and understanding of mathematical concepts. A function’s graph is frequently a useful method of representing the relationship of the function models, and manipulating a mathematical expression for a function can shed light on the function’s properties. Many important phenomena can be modeled using functions presented as expressions.