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Chapter 6: Problem 14

Let \(f(x)=4-\cos x\) (a) Sketch the graph of \(f\) on the interval \([-3 \pi, 3 \pi]\). (b) What is the range of \(f ?\) (c) What is the amplitude of \(f ?\) (d) What is the period of \(f ?\)

Short Answer

Expert verified
(a) Sketch the graph of f(x) = 4 - cos(x) on the interval \([-3\pi, 3\pi]\), which looks like a standard cosine wave shifted upward by 1 unit. (b) The range of f(x) is [3, 5]. (c) The amplitude of f(x) is 1. (d) The period of f(x) is \(2\pi\).

Step by step solution

01

(a) Sketch the graph of f(x)

To sketch the graph of f(x) = 4 - cos(x), remember the basic graph of cos(x). It is a wave oscillating between values -1 and 1, with a period of \(2\pi\) and amplitude 1. Now, to transform the cos(x) function into f(x), subtract 1, which means the wave now oscillates from 3 to 5, instead of -1 and 1. The transformation of 1, or upward shift, has no effect on the period or amplitude, only on the range. To sketch the graph of f(x) on the interval \([-3\pi, 3\pi]\), simply draw the standard cosine wave but shifted upward by 1 unit. The period and amplitude of the function remain the same as the standard cosine function.
02

(b) The range of f(x)

The range of the function f(x) = 4 - cos(x) can be found by looking at its maximum and minimum values. Since its basic trigonometric function is cos(x), which has a maximum value of 1 and a minimum value of -1, we can find the maximum value of f(x) when cos(x) = 1 and the minimum value of f(x) when cos(x) = -1. So, Max(f(x)) = 4 - 1 = 3 Min(f(x)) = 4 - (-1) = 5 Thus, the range of f(x) = 4 - cos(x) is [3, 5].
03

(c) The amplitude of f(x)

The amplitude of a periodic function is half the distance between its maximum and minimum values. In this case, the range of f(x) = 4 - cos(x) is [3, 5]. Hence, Amplitude = (5 - 3) / 2 = 1 Therefore, the amplitude of f(x) = 4 - cos(x) is 1.
04

(d) The period of f(x)

In this case, since our function f(x) is just a constant subtracted from the cosine function, there is no change in its period. So, the period of f(x) = 4 - cos(x) is the same as the period of the cosine function. The period of cos(x) is \(2\pi\). The period of f(x) = 4 - cos(x) is\( 2\pi\).

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