91Ó°ÊÓ

The graph of the inverse cosine, or arccos function, provides a visual representation of how the angle of a cosine function corresponds to its ratio within a right-angle triangle. When sketching the \(\arccos(x)\) graph, it's essential to recognize that it mirrors the \(\cos(x)\) function over the line \(y=x\).

The standard \(\arccos(x)\) function has a domain of \( [-1,1] \) and a range of \( [0, \pi] \). However, when the function is adjusted to \(\arccos(t+2)\), like in the given exercise, the domain and range shift accordingly. For the function \(g(t)=\arccos(t+2)\), the horizontal shift of \(2\) units to the left moves the domain to \( -3 \leq t \leq -1 \).

To effectively sketch the graph, plot the starting point at \( (-3, \pi) \) and the ending point at \( (-1, 0) \). Remember, since \(\arccos\) is a decreasing function, the graph will slope downwards from left to right, forming a part of the shape of a semi-circle.

Understanding the domain and range is crucial when dealing with trigonometric functions, including their inverses. The domain of a function consists of all the input values (x-values) for which the function is defined, while the range represents all the possible output values (y-values).

For the primary trigonometric functions

• The sine and cosine functions have a domain of \( (-\infty, \infty) \) and a range of \( [-1,1] \)
• The tangent function has a domain of all real numbers except where \( \cos(x) = 0 \), and its range is \( (-\infty, \infty) \)

Their inverse functions, including the inverse cosine (or arccos), have domains and ranges that are swapped. The inverse trigonometric functions are restricted in order to have only one value for each input, ensuring they are actual functions. For example, the standard \(\arccos(x)\) function, which we are examining, is restricted to a domain of \( -1 \leq x \leq 1 \) and a range of \( 0 \leq y \leq \pi \) to maintain its function status.

The end behavior of the \(\arccos\) function refers to how the function behaves as it approaches the boundaries of its domain. For the standard \(\arccos(x)\) function, as \(x\) approaches \(1\) from the left, the value of \(\arccos(x)\) approaches \(0\). Conversely, as \(x\) approaches \( -1 \) from the right, the value of \(\arccos(x)\) approaches \(\pi\).

For the modified \(\arccos(t+2)\) function in our exercise, the end behavior shifts with the domain. As \(t\) approaches the left endpoint of the domain, which is \( -3 \), the \(\arccos(t+2)\) approaches the upper end of its range, \(\pi\). At the right endpoint, \( -1 \), the function approaches the lower end of its range, \(0\). This behavior forms part of the distinctive 'n' shape that is characteristic of the inverse cosine function graph. When sketching, you would begin at the upper left-hand corner of the graph, \( (-3, \pi) \) and draw a smooth curve to the lower right-hand corner, \( (-1, 0) \) in a consistent and continuously decreasing manner.