\( \frac{2x+1}{x-3}.\frac{3x}{p(x)}=\frac{6x^{2}+3x}{x^{2}-2x-3} \)

3. Seja \( T: E \rightarrow F \) uma transformação linear. Prove que, \( N(T)=\left\{0_{E}\right\} \) se, e somente se, \( T \) é INJETIVA. 4. Sejam \( S, T: E \rightarrow E \) transformaçôes lineares. Se \( N(T)=N(S)=\left\{0_{E}\right\} \) mostre que \( N(T \circ S)=\left\{0_{E}\right\} \), onde \( T \circ \) é a transformaçâo linear composta dada por \( (T \circ S)(u)=T(S(u)) \), com \( u \in E \). 5. Seja \( T: \mathbb{R}^{2} \rightarrow \mathbb{R} \) dada por \( T(x, y)=3 x+4 y \). Mostre que \( T \) é uma transformação linear.

1. What is the domain of the function?

In horse racing, a trifecta is a bet that the first three finishers in a race are selected, and they are selected in the correct order. Does a trifecta involve combinations or permutations? Explain. Choose the correct answer below. A. Because the order of the first three finishers does not make a difference, the trifecta involves permutations. B. Because the order of the first three finishers does make a difference, the trifecta involves combinations. C. Because the order of the first three finishers does not make a difference, the trifecta involves combinations. D. Because the order of the first three finishers does make a difference, the trifecta involves permutations.

\( \left. \begin{array} { l } { 21 } \\ { \frac { 2600 } { 9 } } \\ { 220 } \end{array} \right. \)

3. Пусть \( \mathbb{Q}:=(\mathbb{Z} \times \mathbb{N} \backslash\{0\}) / \sim \), где \( (a, b) \sim(c, d) \leftrightarrow a d=b c,[a, b] \tilde{+}[c, d]:= \) \( [a d+b c, b d],[a, b] \sim[c, d]:=[a c, b d],[a, b] \leq[c, d] \leftrightarrow a d \leq b c, \tilde{0}:=[0,1] \), \( \tilde{1}:=[1,1] \). \( \quad \) Докажите, что: \( \sim- \) эквивалентность на \( \mathbb{Q} ; \) определения \( \tilde{+}, \tilde{s}, \tilde{\leq} \) кор- ректны; \( (\mathbb{Q} ; \tilde{+}, \tilde{\leq}, \tilde{0}, \tilde{1}) \) есть упорядоченное поле (т.е. упорядоченное коль- цо, в котором \( \forall x \neq 0 \exists y(x \cdot y=1) \) ) такое, что любой элемент является отношнием целого числа и положительного целого числа.