Functions and Their Graphs

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Chapter 3 : Functions and Their Graphs

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

    Thank You from Kimavi arrow_upward

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