# 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.
Solution to Example 1x 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 , + ∞) Find the domain of function f defined by
Matched Problem 1:Answers to matched problems 1,2,3 and 4
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
values.realSolve the above linear inequality x >= 4 The domain, in interval notation, is given by [4 , +∞) Find the domain of function f defined by:
Matched Problem 2:
Solution to Example 3The 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]
^{2} 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 x ^{2} - 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.
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