Question:

Determining period, domain, and range of functions?

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Let y= -2 cos(4x-pi) -5

Determine the period, domain, and range of the function.

Any help on this would be appreciated. Please explain completely so that I fully understand and can do similar problems on my own. Thanks so much and 10 points to a best answer.

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  1. The domain is easy since is  cosine.  

    modifications of y=sin x and y= cos x all have

    (-∞,∞) also known as the entire real number set as their

    domain.

    Now y= -2 cos (4x - π) - 5 is the same as

    y= -2 cos [4(x - π/4)] - 5

    and that has an amplitude of 2

    a period of 2π/4 calculated by dividing basic period

    2π of the general y=cos x function by coefficient 4 inside

    the argument of the cosine operator.

    the transformation where I factored 4 out turns out not needed for this problem, since we don't have to graph

    a representative period.  

    However the range is calculated by looking at the y's and

    the amplitude.  

    The center of the y=cos x  graph is y=0 .

    Since we subtracted 5, the center is going to be the

    horizontal line  y= -5 .   Now since the amplitude is 2 which

    is the absolute value of the number multiplied by the cosine function,   the range is going to vary from -5+2 down to

    -5-2 which is from -3 to -7.    Using interval notation, we

    place the smallest number first so the range is [-7,-3]

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