Piece-wise functions are used to represent functions that have inputs in different range.
The piece-wise function is:
[tex]y =\left[\begin{array}{ccc} 4+ 8x,&0 \le x \le 20\\54+ 5.5x,&21 \le x \le 50\end{array}\right[/tex]
Let the number of packages be x, and the total cost of the package be y
We have:
[tex]Base = 4.00[/tex]
[tex]R_1 = 8.00[/tex] ---rate per package for the first 20
[tex]R_2 = 5.50[/tex] ---rate per package in excess of 20
For the first 20, we have:
[tex]y = Base + R_1 \times x[/tex]
[tex]y = 4+ 8\times x[/tex]
[tex]y = 4+ 8x[/tex]
Where
[tex]0 \le x \le 20[/tex] i.e. from 0 to 20 packages
For packages above 20 but less than 50 (the maximum)
[tex]y = Base + R_1 \times 20 + R_2 \times (x - 20)[/tex]
[tex]y = 4 + 8 \times 20 + 5.5 \times (x - 20)[/tex]
[tex]y = 4 + 160 + 5.5x - 110[/tex]
Collect like terms
[tex]y = 4 + 160 - 110+ 5.5x[/tex]
[tex]y = 54+ 5.5x[/tex]
Where
[tex]21 \le x \le 50[/tex] i.e. from 21 to 50 packages
So, the piece-wise function is:
[tex]y =\left[\begin{array}{ccc} 4+ 8x,&0 \le x \le 20\\54+ 5.5x,&21 \le x \le 50\end{array}\right[/tex]
Read more about piece-wise functions at:
https://brainly.com/question/6111304
