What Is the Domain and Range of y=cos(3(x - 45°)) +2?

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SUMMARY

The domain of the function y=cos(3(x - 45°)) + 2 for one cycle is defined as the interval 0 ≤ 3(x - 45°) ≤ 360°, which translates to the values of x being in the range of -45° ≤ x ≤ 90°. The range of the function is determined by the cosine function's output, which oscillates between -1 and 1. Therefore, the range of y=cos(3(x - 45°)) + 2 is 1 ≤ y ≤ 3. Graphing the function provides a visual representation of these values.

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State the domain and range for one cycle of y=cos(3(x - 45°)) +2 Show your work.
 
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mathuravasant said:
State the domain and range for one cycle of y=cos(3(x - 45°)) +2 Show your work.
Please show us what you have tried and exactly where you are stuck.
We can't help you if we don't where you are stuck.
 
like how do you find domain and range off from that given equation I just don't know what to do
 
mathuravasant said:
like how do you find domain and range off from that given equation I just don't know what to do
You are having a great deal of trouble with these. I suspect the biggest cause isn't in the problems but in the definitions. I would suggest having a 3 x 5 card (or some other modern equivalent) stating what the domain, range, frequency, wave number, horizontal shift (or phase), and vertical shift are for each.

[math]y = cos( 3(x - 45) ) + 2[/math].

What is the domain? It's the interval on the x-axis that the function is defined on. So one cycle is 360 degrees. Thus
[math]0 \leq 3(x - 45) \leq 360[/math]
So what are the possible values for x? [math]0 \leq 3(x - 45)[/math] to [math]3(x - 45) \leq 360[/math]

What is the range? It's the interval on the y-axis that the function takes on over the domain. So for the sake of argument let's say that the domain is [math]-45 \leq x \leq 90[/math]. (Mind you, it isn't.) Then what is the range of cosine? It's best to graph this one and take a look since cosine "waves" so graph [math]y = cos( 3(x - 45) ) + 2[/math] and find the biggest change in y value for [math]-45 \leq x \leq 90[/math].

Let us know how it goes.

-Dan
 

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