Preview Activity 4.3.1.
Consider the plot of the standard cosine function in the following figure along with the emphasized portion of the graph on \([0,\pi]\text{.}\)
Let \(g\) be the function whose domain is \(0 \le t \le \pi\) and whose outputs are determined by the rule \(g(t) = \cos(t)\text{.}\) Note well: \(g\) is defined in terms of the cosine function, but because it has a different domain, it is not the cosine function.
(a)
What is the domain of \(g\text{?}\)
(b)
What is the range of \(g\text{?}\)
(c)
Does \(g\) pass the horizontal line test? Why or why not?
(d)
Explain why \(g\) has an inverse function, \(g^{-1}\text{,}\) and state the domain and range of \(g^{-1}\text{.}\)
(e)
We know that \(g(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\text{.}\) What is the exact value of \(g^{-1}(\frac{\sqrt{2}}{2})\text{?}\) How about the exact value of \(g^{-1}(-\frac{\sqrt{2}}{2})\text{?}\)
(f)
Determine the exact values of \(g^{-1}(-\frac{1}{2})\text{,}\) \(g^{-1}(\frac{\sqrt{3}}{2})\text{,}\) \(g^{-1}(0)\text{,}\) and \(g^{-1}(-1)\text{.}\) Use proper notation to label your results.
