The range of a function is the set of values between the maximum and minimum values that a certain function can output for its domain. The domain of a function is all the function input values that will produce a valid output value.
The standard range of #cos(theta) = [-1,1]# and the domain: #(-oo,oo)#. In other words, the maximum value of #cos(theta)# is 1, the minimum value is -1, and all numbers can be plugged into #cos(theta)# and a valid function value will be found. By multiplying #cos(theta)# by #-3#, all values of #cos(theta)# will be inverted because of the negative sign, and all values will be increased by a factor of 3 as well.
#cos(0) = 1# (maximum value)
#-3(cos(0)) = -3 (1) = -3# (new minimum value)
Multiplying #cos(theta)# by a factor of -3 does not affect the set of valid numbers that can be entered into #cos(theta)# where a valid function value can be found, so the domain of #-3cos(theta)# is the same as #cos(theta)#.
