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The arccos function, also known as the inverse cosine function, is a vital part of trigonometry. It is used to determine the angle whose cosine is a given number. As a reminder, the cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. Thus, the arccos function answers the question: "What angle has a cosine of this value?"

The range of the arccos function is limited. It accepts input values between -1 and 1 due to the nature of the cosine function. Its output, or range, is the angle measure from 0 to \(\pi\) radians. This is because the cosine of any angle can only fall within this numerical interval. It starts at the point (1, 0), reaches the highest point at (0, \(\frac{\pi}{2}\)), and ends at (-1, \(\pi\)), forming a smooth, decreasing curve within this domain.

The function is defined as: \[\text{arccos}(x) = y\text{ such that } \cos(y) = x, \text{ for } x \in [-1, 1] \]Understanding the arccos function is crucial as it lays the foundation for further function transformations.

Function transformations allow us to modify the properties of a function's graph, enabling shifts, stretches, or reflections. When dealing with transformations of the arccos function, it’s essential to consider both vertical and horizontal changes.

In the exercise examined, we consider the function \(f(x) = \text{arccos}\left(\frac{x}{4}\right)\). Here, the argument of arccos is modified from \(x\) to \(\frac{x}{4}\). This indicates a horizontal stretch of the graph by a factor of 4. To understand this, remember that horizontal transformations are the inverse of what may seem intuitive: a multiplication inside the function division implies a stretch.

The new graph is defined for \(x\) values ranging from -4 to 4, expanding the original interval of -1 to 1. This adjustment aligns with the transformation applied to the function’s argument while preserving the function’s core characteristic as an inverse trigonometric function. Function transformations, such as this, are powerful tools for reshaping and reinterpreting graph behaviors.

Sketching graphs offers a visual perspective to understand and analyze functions, making it easier to grasp their behavior and transformations.

For the transformed function \(f(x) = \text{arccos}\left(\frac{x}{4}\right)\), it's important to set key points first. Begin by determining the new critical points based on the transformation:

• Point (1, 0), transforms to (4, 0)
• Point (0, \(\frac{\pi}{2}\)), remains the same
• Point (-1, \(\pi\)), shifts to (-4, \(\pi\))

Next, draw a smooth, decreasing curve that connects these points meticulously, making sure it remains within the x-range of [-4, 4].
It's crucial to maintain a visually balanced graph. Ensure there’s no crossover beyond this defined interval as the function does not provide valid results outside of these boundaries. Graph sketching is not just about plotting points; it’s about understanding and visually representing how the transformed function behaves, offering deeper insights.