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Chapter 4: Problem 20

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitude, period, and shifts. $$\begin{array}{l} f(x)=\cos x \\ g(x)=\cos (x+\pi) \end{array}$$

Short Answer

Expert verified
The amplitude and period of \(f(x)\) and \(g(x)\) are the same (1 and \(2\pi\) respectively). However, \(g(x)\) is shifted to the left by \(\pi\) units compared to \(f(x)\).

Step by step solution

01

Understand the Function f(x)

Start with the first function \(f(x) = \cos x\). Cosine is a standard wave function which oscillates between -1 and 1. It has a period of \(2\pi\), meaning it repeats itself every \(2\pi\) units. The amplitude is 1. There is no shift here, as the cosine function is in its standard form.
02

Understand the Function g(x)

Next, consider the function \(g(x) = \cos (x+\pi)\). This function is also a cosine wave, but it is shifted to the left by \(\pi\) units. This is because \(x+\pi\) effectively means that wherever x would normally take the value 0 (at the peak of the wave), it now takes the value -\(\pi\), and hence the entire wave is shifted to the left by \(\pi\) units. The amplitude and period remain the same as in function \(f(x)\), as no changes have been made to the oscillation or frequency.
03

Compare the Functions

Comparing \(f(x)\) and \(g(x)\), we can see that the amplitude is the same for both functions, at 1. The period is also the same for both functions, at \(2\pi\). However, there is a shift in \(g(x)\) compared to \(f(x)\), specifically a left shift by \(\pi\) units.

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