Question

Which of the following are functions? If f is not a function explain why. \text { i. } f: \mathbb{R} \rightarrow \mathbb{R} \text { with } f(x)=\frac{1}{1-x^{2}} \text { ii. } f: \mathbb{Z} \rightarrow \mathbb{Z} \text { with } f(x)=\frac{x}{2} \text { iii. } f: \mathbb{R} \rightarrow \mathbb{R} \text { with } f(x)=\ln (x) \text { Let } f: \mathbb{R} \rightarrow \mathbb{R} \text { and } g: \mathbb{R} \rightarrow \mathbb{R} \text { with } f(x)=x+2 \text { and } g(x)=-x \text {. Find } g \circ f,(g \circ f)^{-1}, f^{-1} \text { and } g^{-1} \text { Let } f: \mathbb{R}^{*} \rightarrow \mathbb{R} \text { with } f(x)={ }^{x} 1 \text {, where } \mathbb{R}^{*} \text { is the set of all real numbers } different from 0. i. Determine whether or not f is a one to one function ii. Determine whether or not f is an onto function \text { d) Given a function } F: \mathcal{P}(\{a, b, c\}) \rightarrow \mathbb{Z} \text { is defined by } F(A)=|A| \text { for all } A \in \mathcal{P}(\{a, b, c\}) i. Is a one-to-one function? Prove or give a counter-example. ii. Is F an onto function? Prove or give a counter-example. \text { Let } f: A \rightarrow B \text { and } g: B \rightarrow C \text { be functions. Prove that if } g \circ f \text { is one-to-one }

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