Find the domain of the graph of a function; examples with solutions are presented. First, the graphical meaning of the concept of the domain of a function is explained.
Domain of a Graph of a Function
The implied domain of a function \( f \) is the set of all values of \( x \) for which \( f(x) \) is defined and real. The graph of a function \( f \) is the set of all points \( (x, f(x)) \). Hence, for a function \( f \) defined by its graph, the implied domain of \( f \) is the set of all the real values \( x \) along the \( x \)-axis for which there is a point on the given graph.
As an example, there are points on the graph below at \( x = -3, -2.5, -2, -0.5, 2.5, 3, 3.2, 4 \). These values and many other values of \( x \) are included in the domain of \( f \).
There are no points on the graph at \( x = -1 \) (open circle on the graph), \( 0.5 \), \( 1 \), \( 1.5 \), \( 2 \) (open circle). These values and other values of \( x \) are not included in the domain of \( f \).
With these ideas and definitions, we will now solve examples where the entire domain of a given graph is found.
Domain of a Graph: Examples with Detailed Solutions
Example 1
Find the domain of the graph of the function shown below and write it in both interval and inequality notations.
Solution:
The graph starts at \( x = -4 \) and ends at \( x = 6 \). For all \( x \) between \( -4 \) and \( 6 \), there are points on the graph. Hence the domain, in interval notation, is written as:
\[ [-4, 6] \]In inequality notation, the domain is written as:
\[ -4 \leq x \leq 6 \]Note that we use closed brackets because \( -4 \) and \( 6 \) are included in the domain, which is indicated by the closed circles at \( x = -4 \) and \( x = 6 \).
Example 2
What is the domain, in interval notation, of the graph of the function shown below?
Solution:
The graph starts at \( x = -4 \) and ends at \( x = 4 \). There are points on the graph for all values of \( x \) between \( -4 \) and \( 4 \), including at \( -4 \) and \( 4 \). Hence the domain, in interval notation, is written as:
\[ [-4, 4] \]Example 3
What is the domain of the graph of the function?
Solution:
The graph starts at \( x = -8 \) and ends at \( x = 8 \). The graph is defined for all \( x \) between \( -8 \) and \( 8 \). We include \( -8 \) and \( 8 \) because of the closed circles at the endpoints. Hence the domain, in interval notation, is written as:
\[ [-8, 8] \]Example 4
Find the domain of the graph of the function shown below.
Solution:
The graph starts at \( x = -4 \) and ends at \( x = 6 \). The graph is defined for all \( x \) between \( -4 \) and \( 6 \). The interval is closed at \( -4 \) and \( 6 \) due to the closed circles. Hence the domain, in interval notation, is written as:
\[ [-4, 6] \]Example 5
Write the domain of the graph of the function shown below in interval and inequality notations.
Solution:
The graph starts at values of \( x > -4 \) and ends at values of \( x < 4 \). \( x = -4 \) and \( x = 4 \) are not included in the domain because of the open circles at these values. Hence the domain, in interval notation, is written as:
\[ (-4, 4) \]In inequality notation, the same domain is given by:
\[ -4 < x < 4 \]Strict inequality signs are used because the endpoints are not included.
Example 6
Write the domain of the graph of the function shown below in interval notation.
Solution:
The graph starts at \( x = -8 \) and spans across intervals with breaks. The open circles at \( x = -4 \), \( x = -2 \), and \( x = 2 \) indicate that these values are not included in the domain. Hence the domain, in interval notation, is written as:
\[ [-8, -4) \cup (-4, -2) \cup (-2, 2) \]Example 7
Write the domain of the graph of the function shown below in inequality and interval notations.
Solution:
The graph starts at \( x = -4 \) (closed circle) and ends at \( x < 2 \) (open circle). In inequality notation, the domain is written as:
\[ -4 \leq x < 2 \]In interval notation, the domain is given by:
\[ [-4, 2) \]Example 8
Write the domain of the graph of the function shown below using interval notation.
Solution:
The graph is made up of three parts. The left part is defined for all values of \( x \) between \( -4 \) and \( -2 \). The center part is defined for \( x > 0 \) up to \( x \leq 4 \). The right part is defined for \( x > 6 \) up to \( x \leq 8 \). The domain is written as a union of three intervals:
\[ [-4, -2] \cup (0, 4] \cup (6, 8] \]