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.

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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.

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.


Example 1: Find the domain of function f defined by

f (x) = 1 / ( x - 1)

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

    (-infinity , 1) U (1 , +infinity)

Matched Problem 1: Find the domain of function f defined by

f (x) = -1 / ( x + 3)

Answers to matched problems 1,2,3 and 4

Example 2: Find the domain of function f defined by

f (x) = sqrt (2x - 8)

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 , +infinity)

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

f (x) = sqrt (-x + 9)

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

f (x) = sqrt( -x) / [(x - 3) (x + 5)]

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
    (-infinity , -5) U ( -5 , 0]

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

f (x) = sqrt( -x + 2) / [(x + 1) (x + 9)]

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

f (x) = x 2 - 2

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 , +infinity)

  • 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:

f (x) = x 2 + 3

More Find the range of functions, find the domain of a function and mathematics tutorials and problems.





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Updated: 2 April 2013

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