The piecewise function for the dog groomer's charges is defined as follows:
- \( f(x) = 2.5x \) if \( 0 \leq x \leq 20 \)
- \( f(x) = 3.5x \) if \( 20 < x \leq 50 \)
- \( f(x) = 5x \) if \( x > 50 \)
The first piece of the function, \( f(x) = 2.5x \), is valid for dogs weighing between 0 and 20 pounds, inclusive.
The last point graphed on this piece of the function occurs at the endpoint of the interval \( x = 20 \). To find the corresponding charge at this point, we can evaluate the function:
\[ f(20) = 2.5(20) = 50 \]
Thus, the last point graphed on the first piece of the function is \( (20, 50) \).