To define the range of Darien's earnings for walking dogs, let's first clarify the meaning of the variables in the equation \( f(w) = 15w \):
- \( f(w) \) represents the total earnings Darien makes from walking dogs.
- \( w \) represents the number of times he walks the dog.
Given that Darien earns $15 for each walk, the values of \( w \) (the number of walks he can take) must be a non-negative integer since he cannot walk the dog a negative number of times or a fractional amount of times in this context.
Therefore, the real-world constraints give us the following inequalities for \( w \):
- \( w \geq 0 \) (he can walk the dog zero times or more)
- \( w \) can be any whole number depending on how many times he chooses to walk the dog.
Using this information, we can express the range of his earnings in terms of \( f(w) \):
The minimum amount he can earn is $0, which occurs when \( w = 0 \). As \( w \) increases, his total earnings \( f(w) \) can also increase indefinitely, depending on how many times he walks the dog.
Thus, the inequality that shows the range of his earnings can be written as:
\[ f(w) \geq 0 \]
By substituting \( f(w) = 15w \), we can also express it as:
\[ 15w \geq 0 \]
From this, we conclude that his total earnings \( f(w) \) can range from $0 to any positive amount depending on the number of walks he completes. Hence, the range of Darien's earnings can be expressed as \( f(w) \geq 0 \).