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Chapter 6: Problem 27

What is the range of the function \(2+\cos x ?\)

Short Answer

Expert verified
The range of the function \(2 + \cos x\) is \(1 \le 2 + \cos x \le 3\).

Step by step solution

01

Identify the original function and its range

The original function given is \(2 + \cos x\), which is based on the cosine function. The range of the cosine function, \(\cos x\) is \(-1 \le \cos x \le 1\), meaning cosine of any angle x will always have a value between -1 and 1 (inclusive).
02

Add 2 to the range of the cosine function

To determine the range of the new function \(2 + \cos x\), we will add 2 to each part of the range of the cosine function: \[-1 + 2 \le \cos x + 2 \le 1 + 2\]. After calculating, we get: \[1 \le \cos x + 2 \le 3\].
03

Write the answer

The range of the function \(2 + \cos x\) is \(1 \le 2 + \cos x \le 3\), which means for any value of x, the value of the function will always be between 1 and 3 (inclusive).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Cosine Function
The cosine function, represented as \( \cos x \), is a fundamental trigonometric function. It's crucial in understanding periodic behaviors in math and the sciences. This function maps an angle \( x \) to a value between -1 and 1. The cosine wave is periodic and symmetric about the x-axis, oscillating smoothly through these values.
  • The maximum value is 1, occurring at multiples of the full cycle (e.g., 0, \( 2\pi \), etc.)
  • The minimum is -1 at odd multiples of half the cycle (e.g., \( \pi \), \( 3\pi \), etc.)
  • It has a period of \( 2\pi \), meaning it repeats the same values every \( 2\pi \) units.
This predictable pattern makes cosine and its transformations very useful in modeling waves and circular motions.
What is the Range of a Function?
The range of a function is the set of all possible output values it can produce. This is linked directly to the behavior of the function's formula and any transformations it may undergo.
For a simple function like \( \cos x \), the range is straightforward:
  • The function stays within \(-1 \leq \cos x \leq 1\).
When functions undergo transformations, such as vertical shifts, the range changes accordingly.
For example, in the function \( 2 + \cos x \), we adjust the entire range by adding 2 to every value. This shifts the original range of \( \cos x \) from \([-1, 1]\) to \([1, 3]\). Each cosine value is offset by 2, increasing the minimum and maximum bounds.
Understanding Function Transformation
Function transformation involves altering a function's graph in specific ways, such as shifting, stretching, or reflecting it. This helps model real-world phenomena using mathematical functions.
In the exercise, the function \( 2 + \cos x \) is a transformation of \( \cos x \):
  • Vertical Shift: Adding 2 to \( \cos x \) shifts the entire graph upward by 2 units. This affects the range and position on the y-axis.
Unlike stretching or reflection, a vertical shift does not change the period or amplitude but simply repositions the graph.
Recognizing these transformations is key to analyzing how functions behave and predicting their output values. It's like moving a wave up to reflect a change in baseline, without altering its spread or height.

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Most popular questions from this chapter

Convert the rectangular coordinates given for each point to polar coordinates \(r\) and \(\theta .\) Use radians, and always choose the angle to be in the interval \((-\pi, \pi)\). $$ (-5,5) $$

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=10, \theta=\frac{\pi}{6} $$

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=7, \theta=\frac{\pi}{4} $$

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=12, \theta=\frac{11 \pi}{4} $$

What is the period of the function \(7 \cos \left(\frac{\pi}{2} x+\frac{6 \pi}{5}\right) ?\)
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