Question

a rectangle has a length represented by the function ( f(x)=x ) and a width represented by the function ( g(x)=x + 3 ) where both functions are in feet. calculate ( (f cdot g)(x) ) and state what it represents. (1 point) ( (f cdot g)(x)=x^{2}+3 x ); it represents the perimeter of the rectangle in feet. ( (f cdot g)(x)=4 x + 6 ); it represents the perimeter of the rectangle in feet. ( (f cdot g)(x)=x^{2}+3 x ); it represents the area of the rectangle in square feet. ( (f cdot g)(x)=4 x + 6 ); it represents the area of the rectangle in square feet.

Explanation:

Step1: Recall the formula for the area of a rectangle

The area of a rectangle is \(A = \text{length}\times\text{width}\). Given \(f(x)=x\) (width) and \(g(x)=x + 3\) (length).

Step2: Calculate \((f\cdot g)(x)\)

By the definition of the product of two functions \((f\cdot g)(x)=f(x)\cdot g(x)\). Substitute \(f(x)=x\) and \(g(x)=x + 3\) into the formula: \((f\cdot g)(x)=x(x + 3)\).
Using the distributive property \(a(b + c)=ab+ac\) (where \(a = x\), \(b=x\), \(c = 3\)), we get \((f\cdot g)(x)=x^{2}+3x\).

Answer:

\((f\cdot g)(x)=x^{2}+3x\); It represents the area of the rectangle in square feet.