Question

a cable company wants to provide cell phone service for residents on an island. the function for the cost of laying the cable from the island to the mainland is ( c(x)=6,500sqrt{x^{2}+4} ), where ( x ) represents the length of the cable in meters. what are the domain and range of the function?

Explanation:

Step1: Determine the domain

Since \(x\) represents the length of the cable, \(x\geq0\). So the domain is \(x\geq0\) (all non - negative real numbers).

Step2: Determine the range

First, consider the function \(y = \sqrt{x^{2}+4}\). When \(x = 0\), \(y=\sqrt{0 + 4}=2\). As \(x\) increases (\(x\geq0\)), \(y=\sqrt{x^{2}+4}\) also increases.
The function \(C(x)=6500\sqrt{x^{2}+4}\). When \(x = 0\), \(C(0)=6500\times2 = 13000\). As \(x\) increases (\(x\geq0\)), \(C(x)\) increases. So the range is \(C(x)\geq13000\).

Answer:

Domain: \(x\geq0\) (all non - negative real numbers). Range: \(C(x)\geq13000\)