If the cosine function \( g(x) = \cos(x) \) is horizontally shifted to the right by \( \frac{\pi}{4} \), what is the new function?

When you horizontally shift the cosine function \( g(x) = \cos(x) \) to the right by \( \frac{\pi}{4} \), you subtract \( \frac{\pi}{4} \) from the input of the function. Thus, the new function becomes \( g(x) = \cos\left(x - \frac{\pi}{4}\right) \). This means that every point on the cosine curve moves to the right by \( \frac{\pi}{4} \), maintaining the same amplitude and shape but altering its starting point on the x-axis. This shift doesn't change the period of the function, which remains \( 2\pi \). So you can still expect the same repeating pattern, just starting at a different point. Remember that shifting horizontally doesn’t affect the vertical scale, so get ready to groove with that new phase!