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

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Discussion Overview

The discussion centers on determining the domain and range of the function y=cos(3(x - 45°)) + 2 for one cycle. Participants explore the definitions and calculations involved in finding these values, with a focus on both theoretical understanding and practical application.

Discussion Character

  • Homework-related
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants express confusion about how to find the domain and range from the given equation.
  • One participant suggests that understanding the definitions of domain, range, frequency, wave number, horizontal shift, and vertical shift is crucial for solving the problem.
  • Another participant proposes that the domain can be determined by setting up the inequality 0 ≤ 3(x - 45) ≤ 360, although they acknowledge that the suggested domain of -45 ≤ x ≤ 90 may not be accurate.
  • There is a suggestion to graph the function to better understand the range, indicating that visual representation may aid in finding the maximum and minimum y-values.

Areas of Agreement / Disagreement

Participants generally agree on the need to clarify definitions and explore the function graphically, but there is no consensus on the correct domain and range values, and multiple interpretations of the problem remain.

Contextual Notes

There are unresolved assumptions regarding the definitions of domain and range, and the suggested domain may not accurately reflect the function's behavior. The discussion also highlights the importance of visual aids in understanding periodic functions.

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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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