Identity Function

The identity function returns the same value as the input. It is defined as:
f(x) = x
The input values are returned as output, meaning the domain and range are identical. It is a linear function with a slope of 1 and a y-intercept of 0. The graph of an identity function is a straight line that passes through the origin (0, 0) and forms a line at a 45-degree angle to both axes in the Cartesian plane. We get a diagonal line where every point lies on the line y = x.

For example, if f(x) = x, then:

f(2) = 2f(-2) = -2f(0) = 0

The identity function takes a number and gives back the same number, and is the simplest type of function. It just says “What you give is what you get”, which means what we give as input is what we get as output. 

Domain: Domain refers to all the input values of the function. For the identity function, we can input any real number like 1, -2, 7, 0, etc. Thus, the domain includes all real numbers (ℝ).

Range: The range is the set of output values of the function. In the identity function, the output is always the same as the input; therefore, the range is also all real numbers (ℝ).

Inverse: The inverse of the identity function is itself, as it returns the input unchanged. So, applying the identity function again still gives the same result.

To draw the graph of an identity function, take the values of x and find the values for y using the identity function. The graph will be a straight line that passes through the origin. In this function, the range and domain are the same. 
Example: Choose the values of x as: -4, -2, 0, 2, 4
Now find the values of f(x):

Plot the values of x and their corresponding f(x) values on the graph:

Slope and equation of the identity function:

We use the general form of the straight line to find the slope and the equation of the identity function. 
y = mx + c
Here, m is the slope of the line.
c is the point where the line crosses the y-axis.

Finding the slope: To find the slope, pick any two points from the above table and use the slope formula as: 
m = y2 - y1x2 - x1
Consider the points, (x1, y1) = (-2, -2)
(x2, y2) = (2, 2)
Substitute the values into the formula,
m = (2 - (-2)) / (2 - (-2)) = 4 / 4 = 1

Finding the y-intercept: The y-intercept occurs where x = 0. Using f(x) = x, we get the values as x = 0, y = 0. So, the value of c is also 0.

Now substitute m and c into the line equation:
y = mx + c
y = 1x + 0
y = x
Therefore, the equation of the identity function is y = x.

The properties of the identity function play an important role in graphing and linear algebra. The identity function maps each input to itself, with the properties given below: 

• The identity function is a real-valued linear function.

• It has the same values for both domain and range.

• The graph of the identity function is always a straight line, and the slope of the graph is always 1.

•  The identity function is bijective: it is both one-to-one (each input has a unique output) and onto (all possible outputs are covered).

• Composing the identity function with itself results in the identity function.

• The inverse of the identity function is itself.
   

The derivative of a function tells us how fast the value of the function changes as the input changes. This is called the rate of change. 
Let the identity function, f(x) = x
Use the derivative formula:
 d/dx (x) = 1
So, if f(x) = x, then:
d/dx[f(x)] = d/dx(x) = 1
The derivative of the identity function is 1.

The integral of the identity function helps calculate the area under its graph between two points. It is the reverse process of differentiation. 
Let’s take the identity function: f(x) = x
To find the integral, we use the formula,
∫ x dx = x² / 2 + C
Here, C represents the constant of integration.
The integral of the identity function is: 
f(x) dx = x dx = x²/2 + C

There are many methods to check whether a given function is the identity function. Some methods are:

• Compare the given function with the standard identity function f(x) = x. If all input values map to themselves, it is the identity function.

• Plot the function on a graph.

• Substitute some values of x into the function and check if the output matches the input.
   

In daily life, the identity function is applied wherever an input needs to be transferred exactly as it is, such as a computer transferring data, calculating fixed prices in economics, and measuring unaltered signals in engineering and other technical fields. Some of the real-life applications of the identity function include:

• Computer Science: Identity functions are used in programming and software development to pass data unchanged through a function or a system. In functional programming, the identity function is commonly used as a default function or a placeholder function when no changes need to be made to the input.

• Physics: In classical mechanics and vector analysis, Identity functions are used when the coordinate system does not change, like when we are observing motion from a fixed point.

• Machine Learning: In deep learning models, the identity function is used as a linear activation function for certain types of neurons where no change to the input is required.