Answer:
(a) [tex]l = -w + 21[/tex]
(b) Domain: [tex]0 <w < 21[/tex] (See attachment for graph)
(c) [tex]f(w) = -w + 21[/tex]
Step-by-step explanation:
Given
[tex]2(l + w) = 42[/tex]
[tex]l = length[/tex]
[tex]w = width[/tex]
Solving (a): A function; l in terms of w
All we need to do is make l the subject in [tex]2(l + w) = 42[/tex]
Divide through by 2
[tex]l + w = 21[/tex]
Subtract w from both sides
[tex]l + w - w = 21 - w[/tex]
[tex]l = 21 - w[/tex]
Reorder
[tex]l = -w + 21[/tex]
Solving (b): The graph
In (a), we have:
[tex]l = -w + 21[/tex]
Since l and w are the dimensions of the fence, they can't be less than 1
So, the domain of the function can be [tex]0 <w < 21[/tex]
--------------------------------------------------------------------------------------------------
To check this
When [tex]w = 1[/tex]
[tex]l = -1 + 21[/tex]
[tex]l = 20[/tex]
[tex](w,l) = (1,20)[/tex]
When [tex]w = 20[/tex]
[tex]l = -20 + 21[/tex]
[tex]l = 1[/tex]
[tex](w,l)= (20,1)[/tex]
--------------------------------------------------------------------------------------------------
See attachment for graph
Solving (c): Write l as a function [tex]f(w)[/tex]
In (a), we have:
[tex]l = -w + 21[/tex]
Writing l as a function, we have:
[tex]l = f(w)[/tex]
Substitute [tex]f(w)[/tex] for l in [tex]l = -w + 21[/tex]
[tex]l = -w + 21[/tex] becomes
[tex]f(w) = -w + 21[/tex]
