consider the following parametric equations:\n\n$x = \\cos(\\theta)$ and $y = 3\\cos(\\theta)+1$\n\nstep 2…

Explanation:

Step1: Eliminate the parameter

Given (x = \cos(\theta)) and (y=3\cos(\theta)+1). Substitute (x) into the equation for (y). We get (y = 3x+1). But we know that (- 1\leqslant\cos(\theta)\leqslant1). Since (x=\cos(\theta)), the domain of (x) is based on the range of the cosine function.

Step2: Find the domain

Since (x = \cos(\theta)) and (-1\leqslant\cos(\theta)\leqslant1), the domain of (x) (for the function (y = 3x + 1) with the parameter - based constraint) is ([-1,1]).

Step3: Find the range

We use the function (y=3x + 1) and the domain (x\in[-1,1]). When (x=-1), (y=3\times(-1)+1=-3 + 1=-2). When (x = 1), (y=3\times1+1=3 + 1=4). Since (y = 3x+1) is a linear function ((y=mx + b) with (m = 3>0), it is increasing), the range of (y) is found by evaluating the function at the endpoints of the domain.

Answer:

Domain: ([-1,1]), Range: ([-2,4])