Example 4

The graph of the function extends indefinitely to the left and down, as well as to the right and up, as indicated by the arrows on each end. As such, there are no restrictions on either of the variables.

When there are no restrictions on the variables in a function, the input and output can take on any Real Number values, indicated by the symbol R.

Domain: {\(x | x \in \thinspace \rm{R}\)}
Range: {\(y | y \in \thinspace \rm{R}\)}

Using the language of math, the “|” represents “such that” and the “\(\in \)” represents “element of”. So, the above notation is read as “The domain is \(x\) such that \(x\) is an element of the set of Real Numbers” and “The range is \(y\) such that \(y\) is an element of the set of Real Numbers”.

The function shown begins at the point \((−4, 1)\) and extends indefinitely to the right and up. As such, there are restrictions on both variables.

Domain: {\(x | x \ge −4, \thinspace x \in \rm{R}\)}
Range: {\(y | y \ge 1, \thinspace y \in \rm{R}\)}

This is read as, “The domain is \(x\) such that \(x\) is greater than or equal to \(−4\) and \(x\) is an element of the set of Real Numbers” and “The range is \(y\) such that \(y\) is greater than or equal to \(1\) and \(y\) is an element of the set of Real Numbers”. Quite a mouthful written in a much more compact version!

The notation \(x \in \rm{R}\) is still used to help define the domain. Although \(x\) cannot be less than \(−4\), it can be any Real Number value that is greater than or equal to \(−4\).

Similarly, the notation \(y \in \rm{R}\) is still used to help define the range. Although \(y\) cannot be less than \(1\), it can be any Real Number value that is greater than or equal to \(1\).

The function shown begins at the point \((−1, −1)\) and extends to the right and up to the point \((2, 5)\). As such, there are restrictions on both variables.

Domain: {\(x | −1\le x \le 2, \thinspace x \in \rm{R}\)}
Range: {\(y | −1 \le y \le 5, \thinspace y \in \rm{R}\)} 

This is read as “The domain is \(x\) such that \(x\) is greater than or equal to \(−1\), but less than or equal to \(2\) and \(x\) is an element of the set of Real Numbers” and “The range is \(y\) such that \(y\) is greater than or equal to \(−1\), but less than or equal to \(5\) and \(y\) is an element of the set of Real Numbers”.

The notation \(x \in \rm{R}\) is still used to help define the domain. The value of \(x\) can be any Real Number between, and including, \(−1\) and \(2\).

Similarly, the notation \(y \in \rm{R}\) is still used to help define the range. The value of \(y\) can be any Real Number between, and including, \(−1\) and \(5\).