To identify the function based on the graph, you would need to look for specific characteristics, such as the frequency, amplitude, and general shape of the graph.
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If the graph is oscillating between -1 and 1 with a period of \(\frac{2\pi}{8} = \frac{\pi}{4}\), then it is likely a cosine or sine function with an angular frequency of 8.
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If the function appears to start at its maximum value (when \(x=0\)), it would be a cosine function:
- \(f(x) = \cos(8x)\) or \(f(x) = \cos(4x)\) based on their respective frequencies.
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If the function starts at zero and goes up, it would be a sine function:
- \(f(x) = \sin(8x)\) or \(f(x) = \sin(x8)\) (though mathematically these are equivalent).
Given the options:
- \(f(x) = \cos(8x)\)
- \(f(x) = \sin(8x)\)
- \(f(x) = \sin(x8)\)
- \(f(x) = \cos(4x)\)
Without the actual graph, if you can note its behavior around \(x=0\) and the period, that would help identify the correct function. If the graph is periodic with a shorter wavelength (indicating a higher frequency), it is likely \(f(x) = \cos(8x)\) or \(f(x) = \sin(8x)\).
In conclusion, if the graph looks like a cosine function starting at its peak, then choose \(f(x) = \cos(8x)\). If it's a sine function starting at zero, choose \(f(x) = \sin(8x)\).
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