The domain of a function is the set of all input values for which it is defined.
For example, consider the function represented by the mapping diagram below.
The domain corresponds to the set of values in the "Input" set. Therefore, the domain of this function is
Let's now consider a situation where a function is defined using a table.
What is the domain of the function defined by the following table?
The domain is the set of all input values for which the function is defined. In the table, this corresponds to the set of -values in the first row:
What is the domain of the function $f$ defined by the mapping diagram above?
|
a
|
$\{ -2,-1,0,1,2 \}$ |
|
b
|
$\{ -8,-2,-1,0,1,2,8 \}$ |
|
c
|
$(-2,2)$ |
|
d
|
$(-8,8)$ |
|
e
|
$\{ -8,-1,0,1,8 \}$ |
What is the domain of the function $f$ defined by the following table?
| $x$ | $1$ | $2$ | $3$ | $5$ | $7$ |
| $y$ | $1$ | $4$ | $9$ | $16$ | $25$ |
|
a
|
$\{ 1,4,9,16,25 \}$ |
|
b
|
$\{ 1,2,3,5,7\}$ |
|
c
|
$[1,7]$ |
|
d
|
$(1,7)$ |
|
e
|
$(1,25)$ |
Let's consider the function shown below.
Note the following:
The function is not defined for nor is it defined for
The function has filled circles at and so it is defined at both of these points.
In addition, the function is defined for every value of between and
Therefore, the domain of this function is
We can also express the domain using interval notation:
The square brackets indicate that the endpoints and are included in the domain.
Now consider the function shown below.
Note the following:
The function is not defined for nor is it defined for
The function has open circles at and so it is not defined at either of these points.
However, the function is defined for every value of between and
Therefore, the domain of this function is
which we can write using interval notation as
The parentheses indicate that the endpoints and are excluded from the domain.
It's possible for a function to have one endpoint included in the domain and another endpoint excluded.
For example, consider the function is shown below.
The endpoint is included in the domain, but the endpoint is excluded. Therefore, the domain of is
which we can write using interval notation as
What is the domain of the function shown below?
The function above is defined for values between and
Since there is a closed circle at this point is included in the domain.
Since there is a closed circle at this point is also included in the domain.
Therefore, the domain of this function is
Using interval notation, we can write this as
What is the domain of the function given above?
|
a
|
$x\in[3,4]$ |
|
b
|
$x\in(2,4)$ |
|
c
|
$x\in[2,4]$ |
|
d
|
$x\in(1,4)$ |
|
e
|
$x\in[1,4]$ |
What is the domain of the function $f(x)$ given above?
|
a
|
$x\in (1, 5]$ |
|
b
|
$x\in (1, 4]$ |
|
c
|
$x\in [1, 4]$ |
|
d
|
$x\in [1, 4)$ |
|
e
|
$x\in [1, 5)$ |
Consider the function shown below.
No matter which number we pick as an input, always will return an output. Therefore, we can write the domain of this function as
Let's compare this to the function shown below:
We cannot substitute the value into the function because division by zero is undefined. Therefore, is not part of the domain of However, we can use any other number as input for Therefore, the domain of is
This notation means that any number apart from zero can be used as an input for
What is the domain of the function shown above?
The function above is defined for all values of that are less than
In addition, since there is a closed circle at this point is included in the domain.
Therefore, the domain of this function is which we can write as
What is the domain of the function given above?
|
a
|
$[2,\infty)$ |
|
b
|
$(-\infty, 2]$ |
|
c
|
$(-\infty, 2)$ |
|
d
|
$(2,\infty)$ |
|
e
|
$(-\infty,\infty)$ |
What is the domain of the function shown above?
|
a
|
$x \in \{-3, -2, -1, 0, 1, 2, 3\}$ |
|
b
|
$x = 1$ |
|
c
|
$x \in (-\infty, 1) \cup (1, \infty)$ |
|
d
|
$x \in \{-3, -2, -1, 0, 1, 2, 3 ,4, 5\}$ |
|
e
|
$x \in (-\infty, \infty)$ |