Question

graph the function ( y = 2cosleft(\frac{1}{2}x
ight) ). show at least two cycles. use the graph to determine the domain and range of the function.
use the graphing tool to graph the equation. type pi to insert ( pi ) as needed.
use the graph to determine the domain of ( y = 2cosleft(\frac{1}{2}x
ight) ).
(type your answer in interval notation. use integers or fractions for any numbers in the expression.)
use the graph to determine the range of ( y = 2cosleft(\frac{1}{2}x
ight) ).
(type your answer in interval notation. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the domain of cosine function

The function \(y = A\cos(Bx - C)+D\) is a transformation of the basic cosine function \(y=\cos(x)\). The domain of \(y = \cos(x)\) is all real numbers, \((-\infty,\infty)\). For the function \(y = 2\cos(\frac{1}{2}x)\), since there is no restriction on the value of \(x\) for which the cosine function is defined, the domain is based on the nature of the cosine function.

Step2: Recall the range of cosine function

The range of \(y=\cos(x)\) is \([- 1,1]\). For the function \(y = A\cos(Bx - C)+D\), the range is given by \([D - |A|,D + |A|]\). In the function \(y = 2\cos(\frac{1}{2}x)\), \(A = 2\), \(B=\frac{1}{2}\), \(C = 0\), \(D = 0\). Using the formula for the range of \(y=A\cos(Bx - C)+D\), we substitute \(A = 2\) and \(D = 0\) into \([D - |A|,D + |A|]\).

Answer:

• Domain: \((-\infty,\infty)\)
• Range: \([-2,2]\)