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Functions and Relations

Writing Functions

There are a lot of word problems which require an ability to correctly model a real-world problem with Algebraic functions. The following shows examples on writing and algebraic functions to model real-world situations.
 
The perimeter of a rectangular field is \( 80 meters \). Find the area of the field in terms of its length \( \mathbf{x}. \)
 
Solution:
 
Representations: P = Perimeter; A = Area
Let $P = 2x + 2y$ where $\mathbf{x}$ is the field's length and y is its width.
       $A = xy$
Given: $P = 80$
            $80 = 2x + 2y$
\(40 = x + y \) (divide both sides by 2)
\(40 - x = y \) (solve for y)
Solving for the Area of the given rectangular field, it follows that:
\(A = xy\)
 
\(A = x(40 - x)\) (substitute the derived value of y above)
\(A = 40x - x^2\)
\(A(x) = 40x - x^2 \)

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