f(x)=x

^{2}

f:function name

x:input

x

^{2}:what to output

There are different types of functions in mathematics like

Polynomial Function, Trigonometric Functions, Exponential Functions, Logarithmic, Implicit Function, Inverse Trigonometric Functions signum,

### ABSOLUTE VALUE FUNCTION FORMULA

The function f(x)=|x| is called an absolute value function.This demonstrates that the absolute value function catches the actual value if x holds a value equivalent to or greater than zero, but if x is less than 0 then function picks minus times the actual value of the original.

f(x)=|x| = {x 😡 > 0}

{-x 😡 < 0}

#### PROPERTIES OF Modulus FUNCTION

For any real number x, we have: √x=|x|||x||=|x|

If a and b denote positive real numbers, then: The absolute value of a positive number is positive. The absolute value of a negative number is obtained by ignoring the minus sign. Thus, the modulus function always possesses non-negative values.

### DIFFERENTIATION OF ABSOLUTE VALUE FUNCTION:

Since we know that an absolute value function f(x)=|x| is equal to x if x>0 and-1 if x<0. The derivative of the absolute value function is not defined for x=0. Hence the derivative of absolute value function is x/|x|, x not equal to 0. Absolute value function x is not differentiable at x=0 as the graph of Mod(x) has a sharp point at x=0. Also, the left-hand limit and the right-hand limit are not equal at x=0. Absolute value function problems can be solved by applying modulus to a non-negative number and a negative number always results in the same number.### INTEGRATION OF ABSOLUTE VALUE FUNCTION:

The integral of x depends on the domain in which it is being integrated.For x>0, |x|=x.

For x<0, |x|=-x.

The integration of the modulus function can be clubbed as:

If |x| is integrated into the positive domain, the result obtained is: