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Chapter 4: Problem 69

Graph the functions \(f\) and \(g .\) Use the graphs to make a conjecture about the relationship between the functions. $$f(x)=\sin x+\cos \left(x+\frac{\pi}{2}\right), \quad g(x)=0$$

Short Answer

Expert verified
The function \(f(x)\) will look like regular sine or cosine waves oscillating between -1 and 1, shifted by \(\frac{\pi}{2}\), whereas \(g(x)\) will be a horizontal line along the x-axis. The two functions intersect when \(f(x) = 0\), which are the solutions to the equation \(f(x) = g(x)\). Thus, we can conjecture that the function \(f(x)\) has roots at the points where it intersects with \(g(x)\).

Step by step solution

01

Preliminary Analysis of the Functions

Start by observing the given functions. The function \(f(x)=\sin x+\cos \left(x+\frac{\pi}{2}\right)\) is a sum of a sine and a cosine function, while \(g(x)=0\) is a constant function.
02

Graphing the Functions

Utilize a graphing tool or technique to plot each function. For function \(f(x)\), sin(x) will oscillate between -1 and 1 and cos \left(x+\frac{\pi}{2}\right) will also oscillate between -1 and 1. For function \(g(x)\), it is a horizontal line on the x-axis. As a result, all the y-values will be 0 for function \(g(x)\).
03

Making Conjectures

Following the plots, infer relationships between the two functions. Look closely at the function \(f(x)\) and \(g(x)\) and see where \(f(x)\) intersects with \(g(x)\). These points represent the solutions of the equation \(f(x) = g(x)\).

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