# Chapter 3 : Functions and Their Graphs

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

#### Graph of a Function arrow_upward

• 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.

• #### Linear Function arrow_upward

• 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 arrow_upward

• 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 arrow_upward

• Square Root Function is defined by
• Domain of
• Range of

#### Modulus Function arrow_upward

• 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:
• Domain of f(x) = R
• Range of

#### Signum Function arrow_upward

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

• Domain of sgn (x) = R
• Range of sgn (x) = {-1, 0, 1}

• #### Exponential Functions arrow_upward

• 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 .
•

• The graph of the function is as shown below, which is increasing if  .
• Decreasing if 0 < a < 1

• #### Logarithmic Functions arrow_upward

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

• Properties
•

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

• #### Power Function arrow_upward

• 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:
• Domain of f(x) = R
• Range of
• If n is a positive odd integer
• Example:
• Domain of f(x) = R
• Range of f(x) = R

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