Domain and Range of a Function

A step by step tutorial, with detailed solutions, on how to find the domain and range of real valued functions is presented. First the definitions of these two concepts are presented. A table of domain and range of basic functions might be useful to answer the questions below.

Definition of the Domain of a Function

For a function f defined by an expression with variable x, the implied domain of f is the set of all real numbers variable x can take such that the expression defining the function is real. The domain can also be given explicitly.
also Step by Step Calculator to Find Domain of a Function

>Definition of the Range of a Function

The range of f is the set of all values that the function takes when x takes values in the domain.
Also Step by Step Calculator to Find Range of a Function


Example 1: Find the domain of function f defined by


Solution to Example 1
  • x can take any real number except 1 since x = 1 would make the denominator equal to zero and the division by zero is not allowed in mathematics. Hence the domain in interval notation is given by the set
    (- ∞ , 1) U (1 , + ∞)
    Matched Problem 1: Find the domain of function f defined by

    Answers to matched problems 1,2,3 and 4


    Example 2: Find the domain of function f defined by


    Solution to Example 2


    The expression defining function f contains a square root. The expression under the radical has to satisfy the condition
    2x - 8 >= 0    for the function to take real values.
    Solve the above linear inequality
    x >= 4
    The domain, in interval notation, is given by
    [4 , +∞)

    Matched Problem 2: Find the domain of function f defined by:


    Example 3: Find the domain of function f defined by:


    Solution to Example 3
    The expression defining function f contains a square root. The expression under the radical has to satisfy the condition
    -x ≥ 0
    Which is equivalent to
    x ≤ 0
    The denominator must not be zero, hence x not equal to 3 and x not equal to -5.
    The domain of f is given  by
    (-∞ , - 5) ∪ ( - 5 , 0]

    Matched Problem 3: Find the domain of function f defined by:



    Example 4: Find the range of function f defined by:

    Solution to Example 4

    The domain of this function is the set of all real numbers. The range is the set of values that f(x) takes as x varies. If x is a real number, x2 is either positive or zero. Hence we can write the following:
    x 2 ≥ 0
    Subtract - 2 to both sides to obtain
    x 2  - 2 ≥ - 2
    The last inequality indicates that x2 - 2 takes all values greater that or equal to - 2. The range of f is given by
    [ -2 , +∞)
    A graph of f also helps in interpreting the range of a function. Below is shown the graph of function f given above. Note the lowest point in the graph has a y (= f (x) ) value of - 2.
    Graph of Function in Example 4

    Matched Problem 4: Find the range of function f defined by:


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