# Chapter 3 : Functions and Their Graphs

### Topics covered in this snack-sized chapter:

The graph of the function f(x) is the set of points (x, y) in the xy plane that satisfy the relation y = f(x).
The graph of a function f and the graph of its inverse are symmetric with respect to the line.
If a and b be fixed real numbers, then the Linear Function is defined as y = f(x) = ax + b, where a and b are constants.

Constant function is a linear function of the form y = c, where c is a constant. It is also written as f(x) = c.
The graph of a constant function is a line.
The function that associates to each real number x, this fixed number c, is called a Constant Function.

Square Root Function is defined by
- Domain of

- Range of

The modulus function returns positive value of a variable or an expression. For this reason, this function is also referred as absolute value function.
Modulus Function is given by , where denoted the absolute value of , that is:

- Range of

Signum function is denoted by ,
thus, f (x) = sgn (x).
Where,

Domain of sgn (x) = R
Range of sgn (x) = {-1, 0, 1}
If a is any number such that a > 0 and a 1 then an exponential function is a function in the form, , where a is called the base and x can be any real number.
The Exponential Function with base is the following function from R to .

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The graph of the function is as shown below, which is increasing if .

Decreasing if 0 < a < 1

If a is any number such that a > 0 and a ≠ 1 and x > 0 then, .
Symbolically,

Properties

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The graph of the functions is as shown below, which is increasing if a > 1

Decreasing if

The Power Function is given by .The graph of y = f(x), its domain and range depend on n.
If n is positive even integer

- Example:

- Range of

If n is a positive odd integer

- Example: