The function \( y = 4x + 7 \) is a linear function. Linear functions are defined for all real numbers, meaning that there are no restrictions on the values that \( x \) can take.
Therefore, the domain of the function \( y = 4x + 7 \) is:
\[ \text{Domain} = (-\infty, \infty) \]
This means that \( x \) can be any real number.