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To understand x-intercepts, we need to identify where the graph of the function crosses the x-axis. This point occurs when the output of the function, denoted as \( f(x) \), is zero. For our given function, \( f(x) = \frac{\sin 4x}{2x} \), the task is to find where this fraction equals zero. In rational functions, the fraction equals zero only when its numerator is zero. Therefore, set \( \sin(4x) = 0 \). The general solutions to this equation are \( 4x = n\pi \), where \( n \) is any integer. Solving for \( x \), we get \( x = \frac{n\pi}{4} \).However, we must remember that \( f(x) \) is not defined at \( x = 0 \) due to the denominator. Nonetheless, all multiples of \( \frac{\pi}{4} \), apart from \( x = 0 \), are valid x-intercepts for our function.

Graphing a function involves plotting its behavior on a coordinate plane. For our function, \( f(x) = \frac{\sin 4x}{2x} \), we should focus on key characteristics such as intercepts and symmetry. Since the function is even, it exhibits symmetry about the y-axis. This means that the left side of the graph mirrors the right side. To graph it effectively:

• Begin by plotting the function’s x-intercepts, which occur at \( x = \frac{n\pi}{4} \), excluding \( x = 0 \).
• Be mindful of the undefined point at \( x = 0 \), indicating the presence of a vertical asymptote or removable discontinuity.
• Choose a viewing window that spans from \(-2\pi\) to \(2\pi\) in the x-direction and from \(-1\) to \(1\) in the y-direction.

With these steps, you'll capture the essence and structure of the function’s graph.

Analyzing end behavior helps us understand what happens to the function values as \( x \) moves towards positive or negative infinity. For the function \( f(x) = \frac{\sin 4x}{2x} \), the key is to observe how the values settle as \( |x| \) becomes very large. The limits, \( \lim_{x \to \infty} \frac{\sin 4x}{2x} = 0 \) and \( \lim_{x \to -\infty} \frac{\sin 4x}{2x} = 0 \), confirm the function's outputs (or y-values) approach zero as \( x \) trends towards both infinities.

• This indicates horizontal asymptotic behavior, where the graph flattens and becomes almost horizontal as \( x \) increases or decreases without bound.
• Such behavior is characteristic of functions where the degree of the polynomial in the denominator exceeds that of the numerator.

One interesting aspect of many functions is their discontinuities. For \( f(x) = \frac{\sin 4x}{2x} \), the function is not defined at \( x = 0 \), indicating a discontinuity. However, we can explore further. When approaching discontinuities, mathematicians often use limits to analyze their nature. For this function, as \( x \to 0 \),\[\lim_{x \to 0} \frac{\sin 4x}{2x} = \lim_{x \to 0} 2 \left( \frac{\sin 4x}{4x} \right) = 2 \times 1 = 2,\] which means the function approaches a value of 2.

• In this case, the discontinuity at \( x = 0 \) is removable because, despite the hole at this point, the function smoothly approaches a limit.
• If graphed, the function will appear continuous, except for the point at \( x = 0 \) which technically is omitted in simple plots.

Understanding these discontinuities is crucial for mastering function analysis.