91Ó°ÊÓ

The inverse cosine function, denoted as \( ext{arccos}(x)\), is a function that helps us find the angle whose cosine value is \(x\). This is used when you need to determine angles in various fields such as physics or engineering.
Understanding this concept is crucial because it allows us to work backward from cosine values to angles.

• **Behavior**: The \( ext{arccos}(x)\) function decreases as \(x\) moves from -1 to 1.
• **Angles**: The angle associated ranges from 0 to \( ext{arccos}(-1) = \pi\).

For example, if you know that \(\cos(\theta) = \sqrt{3}/2\), using inverse cosine, you can find that \(\theta = \text{arccos}(\sqrt{3}/2)\), which will give you a specific angle within the range of 0 to \(\pi\). In this exercise, the transformation occurs because of the function \(g(t) = \text{arccos}(t + 2)\).
This implies a horizontal shift that changes the domain as compared to the typical \( ext{arccos}\) function.

Sketching the graph of \(g(t) = \text{arccos}(t+2)\) involves understanding its transformation and behavior. The shift in the function by 2 units to the left significantly affects its domain. To sketch accurately, start by:

• **Identify Key Points**: Calculate points like \((t, g(t))\) by substituting key values of \(t\). For instance, \(g(-3) = \pi\) and \(g(-1) = 0\).
• **Decreasing Nature**: Remember that the graph decreases continuously from the leftmost point to the rightmost point.

To draw the graph:1. Begin at \((-3, \pi)\) since this is one boundary of the domain.2. Draw a smooth, decreasing curve that ends at \((-1, 0)\).
Remember, the function remains in the range \([0, \pi]\), making it essential to ensure your graph doesn't extend beyond these vertical limits.

The terms **domain** and **range** are foundational concepts when dealing with functions like \( ext{arccos}(x)\).- **Domain**: Refers to all possible input values (\(x\) values) for the function. For \( ext{arccos}(x)\), the domain is originally \([-1, 1]\). However, due to the horizontal shift in \(g(t) = \text{arccos}(t+2)\), the domain adjusts to \([-3, -1]\).- **Range**: Represents the set of possible output values (\(y\) values) of the function. The range of \( ext{arccos}(x)\) is always \([0, \pi]\) regardless of horizontal shifts.Understanding these concepts:

• Helps you determine the endpoints of the graph, as seen with the points \((-3, \pi)\) and \((-1, 0)\).
• Ensures the output values stay within physical constraints, such as angles in radians.

Thus, determining the domain and range is a vital step in graph sketching, ensuring precise and accurate representations of transformations of the inverse cosine function.