# 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 a Step by Step Calculator to Find Range of a Function is included in this website.

## Examples with Detailed Solutions

### Example 1

Find the domain of function $$f$$ defined by $f(x) = \dfrac{1}{x-1}$ 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 the set
$$(- \infty, 1) \cup (1, + \infty)$$

### Matched Problem 1

Find the domain of function $$f$$ defined by $f(x) = \dfrac{-1}{x+3}$

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
$$2x - 8 \geq 0$$    for the function to take real values.
Solve the above linear inequality
$$x \geq 4$$
The domain, in interval notation, is given by
$$[4, +\infty)$$

### 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: $f(x) = \dfrac{\sqrt{-x}}{(x-3)(x+5)}$ Solution to Example 3
The expression defining function $$f$$ contains a square root. The expression under the radical has to satisfy the condition
$$-x \geq 0$$
Which is equivalent to
$$x \leq 0$$
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
$$(-\infty, -5) \cup (-5, 0]$$

### Matched Problem 3

Find the domain of function $$f$$ defined by: $f(x) = \dfrac{\sqrt{-x+2}}{(x+1)(x+9)}$

### 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, $$x^2$$ is either positive or zero. Hence we can write the following:
$$x^2 \geq 0$$
Subtract - 2 to both sides to obtain
$$x^2 - 2 \geq -2$$
The last inequality indicates that $$x^2 - 2$$ takes all values greater than or equal to - 2. The range of $$f$$ is given by
$$[-2, +\infty)$$
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.

### Matched Problem 4

Find the range of function $$f$$ defined by: $f(x) = x^2 + 3$