91Ó°ÊÓ

Let \(R_{5}=\\{0,1,2,3,4\\}\) (a) Define \(f: R_{5} \rightarrow R_{5}\) by \(f(x)=x^{2}+4(\) mod 5\()\) for all \(x \in R_{5} .\) Write the inverse of \(f\) as a set of ordered pairs and explain why \(f^{-1}\) is not a function. (b) Define \(g: R_{5} \rightarrow R_{5}\) by \(g(x)=x^{3}+4(\) mod 5\()\) for all \(x \in R_{5} .\) Write the inverse of \(g\) as a set of ordered pairs and explain why \(g^{-1}\) is a function. (c) Is it possible to write a formula for \(g^{-1}(y),\) where \(y \in R_{5} ?\) The answer to this question depends on whether or not is possible to define a cube root of elements of \(R_{5} .\) Recall that for a real number \(x,\) we define the cube root of \(x\) to the real number \(y\) such that \(y^{3}=x .\) That is, $$ y=\sqrt[3]{x} \text { if and only if } y^{3}=x $$ Using this idea, is it possible to define the cube root of each number in \(\mathbb{Z}_{5} ?\) If so, what are \(\sqrt[3]{0}, \sqrt[3]{1}, \sqrt[3]{2}, \sqrt[3]{3},\) and \(\sqrt[3]{4}\). (d) Now answer the question posed at the beginning of Part (c). If possible, determine a formula for \(g^{-1}(y)\) where \(g^{-1}: R_{5} \rightarrow R_{5}\)

The Inverse Sine Function. We have seen that in order to obtain an inverse function, it is sometimes necessary to restrict the domain (or the codomain) of a function. (a) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=\sin x\). Explain why the inverse of the function \(f\) is not a function. (A graph may be helpful.) Notice that if we use the ordered pair representation, then the sine function can be represented as $$ f=\\{(x, y) \in \mathbb{R} \times \mathbb{R} \mid y=\sin x\\} $$ If we denote the inverse of the sine function by \(\sin ^{-1}\), then $$ f^{-1}=\\{(y, x) \in \mathbb{R} \times \mathbb{R} \mid y=\sin x\\} $$ Part (14a) proves that \(f^{-1}\) is not a function. However, in previous mathematics courses, we frequently used the "inverse sine function." This is not really the inverse of the sine function as defined in Part (14a) but, rather, it is the inverse of the sine function restricted to the domain \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\). (b) Explain why the function \(F:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow[-1,1]\) defined by \(F(x)=\) \(\sin x\) is a bijection. The inverse of the function in Part ( \(14 \mathrm{~b}\) ) is itself a function and is called the inverse sine function (or sometimes the arcsine function). (c) What is the domain of the inverse sine function? What are the range and codomain of the inverse sine function? Let us now use \(F(x)=\operatorname{Sin}(x)\) to represent the restricted sine function in Part (14b). Therefore, \(F^{-1}(x)=\operatorname{Sin}^{-1}(x)\) can be used to represent the inverse sine function. Observe that $$ F:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow[-1,1] \text { and } F^{-1}:[-1,1] \rightarrow\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$ (d) Using this notation, explain why \(\operatorname{Sin}^{-1} y=x\) if and only if \(\left[y=\sin x\right.\) and \(\left.-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\right]\) \(\operatorname{Sin}\left(\operatorname{Sin}^{-1}(y)\right)=y\) for all \(y \in[-1,1] ;\) and $$ \operatorname{Sin}^{-1}(\operatorname{Sin}(x))=x \text { for all } x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$

Let \(C\) be the set of all real functions that are continuous on the closed interval \([0,1] .\) Define the function \(A: C \rightarrow \mathbb{R}\) as follows: For each \(f \in C,\) $$A(f)=\int_{0}^{1} f(x) d x$$ Is the function \(A\) an injection? Is it a surjection? Justify your conclusions.

Piecewise Defined Functions. We often say that a function is a piecewise defined function if it has different rules for determining the output for different parts of its domain. For example, we can define a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) by giving a rule for calculating \(f(x)\) when \(x \geq 0\) and giving a rule for calculating \(f(x)\) when \(x<0\) as follows: $$f(x)=\left\\{\begin{array}{ll} x^{2}+1, & \text { if } x \geq 0 \\ x-1 & \text { if } x<0 \end{array}\right.$$ (a) Sketch a graph of the function \(f .\) Is the function \(f\) an injection? Is the function \(f\) a surjection? Justify your conclusions. For each of the following functions, determine if the function is an injection and determine if the function is a surjection. Justify all conclusions. (b) \(g:[0,1] \rightarrow(0,1)\) by (c) \(h: \mathbb{Z} \rightarrow\\{0,1\\}\) by $$g(x)=\left\\{\begin{array}{ll} 0.8, & \text { if } x=0 \\ 0.5 x, & \text { if } 0