Activity 1.7.3.
Determine, with justification, whether each of the following functions has an inverse function. For each function that has an inverse function, give two examples of values of the inverse function by writing statements such as “\(s^{-1}(3) = 1\)”.
(a)
The function \(f : S \to S\) given by the table of values below, where \(S = \{0, 1, 2, 3, 4 \}\text{.}\)
| \(x\) | 0 | 1 | 2 | 3 | 4 |
| \(f(x)\) | 1 | 2 | 4 | 3 | 2 |
(b)
The function \(g : S \to S\) given by the table of values below, where \(S = \{0, 1, 2, 3, 4 \}\text{.}\)
| \(x\) | 0 | 1 | 2 | 3 | 4 |
| \(g(x)\) | 4 | 0 | 3 | 1 | 2 |
(c)
The function \(p\) given by \(p(t) = 7 - \frac{3}{5}t\text{.}\) Assume that the domain and codomain of \(p\) are both “all real numbers”.
(d)
The function \(q\) given by \(q(t) = 7 - \frac{3}{5}t^4\text{.}\) Assume that the domain and codomain of \(q\) are both “all real numbers”.
(e)
The functions \(r\) and \(s\) given by the graphs in the figures below. Assume that the graphs show all of the important behavior of the functions and that the apparent trends continue beyond what is pictured.
