Understanding the Identity Function: Definition, Graph, and Properties

Identity Function

The identity function, denoted as f(x) = x, is a basic mathematical function that simply returns the input value as its output

The identity function, denoted as f(x) = x, is a basic mathematical function that simply returns the input value as its output. In other words, whatever value you put into the function, it gives you the same value back.

The graph of the identity function is a straight line diagonal to the Cartesian coordinate system, passing through the origin (0, 0). The slope of this line is 1, indicating that for every unit increase in the x-value, the y-value also increases by the same amount.

Here’s an example to illustrate how the identity function works:

If we have the input value x = 5, the identity function f(x) = x will return the same value 5 as its output. So, f(5) = 5.

Similarly, if we have x = -3, the function will give f(-3) = -3 as the output.

In general, the identity function preserves the values of the input without any alteration. This function is used as a reference point or comparison for other functions, as it provides a baseline for comparison.

Some key properties of the identity function are:

1. It is an injective function, which means that each input value corresponds to a unique output value.
2. It is also a surjective function, as it covers the entire range of real numbers. Every real number is an output of the identity function.
3. It is an example of a linear function, with a constant slope of 1 and a y-intercept of 0.
4. The composition of the identity function with any other function results in the original function. For example, if we compose f(x) = x with another function g(x), we get f(g(x)) = g(x).

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