4. Find the negation of each of the following quantified propositions and determine their truth value on the set of Natural numbers :- a. \( \left(\forall_{x}\right)\left(\forall_{y}\right)\left(x^{2}+y^{2} \neq 1\right) \) b. \( \left(\exists_{y}\right)\left(\forall_{x}\right)\left(x^{2}-y^{2}=4\right) \)

Consider the following. \[ \begin{array}{l}P(x)=\frac{1-5 x}{1-x} \\ \text { (a) State the domain of the function. } \\ \text { all real numbers } x \text { except } x=\frac{1}{5} \\ \text { all real numbers } x \text { except } x=-\frac{1}{5} \\ \text { all real numbers } x \text { except } x=-1 \\ \text { all real numbers } x\end{array} \] all real numbers \( x \) except \( x=1 \)

Find an exponential function of the form \( P(t)=y_{0} e^{\mathrm{Kt}} \) to model the given data. In 1985, the number of female athletes participating in Summer Olympic-Type Games was 550. In 1996, about 3,650 participated in the Summer Olympics-Type games. Assuming that \( P(0)=550 \) and that the exponential model applies, find the value of \( k \) rounded to the hundredths place, and write the function. A. \( k=0.27 ; P(t)=550 e^{0.27 t} \) B. \( k=0.17 ; P(t)=550 e^{0.17 t} \) C. \( k=0.16 ; P(t)=550 e^{0.16 t} \) D. \( k=0.19 ; P(t)=550 e^{0.19 t} \)

Consider the following. \[ f(x)=\frac{3 x^{2}-19 x+20}{3 x^{2}-x-70} \] (a) State the domain of the function. all real numbers \( x \) except \( x=\frac{4}{3}, 5 \) all real numbers \( x \) except \( x=-\frac{5}{3}, 14 \) all real numbers \( x \) except \( x=-\frac{14}{3}, 5 \) all real numbers \( x \) except \( x=-5, \frac{14}{3} \) Identify all intercepts; (If an answer does not exist, enter DNE.) \( x \)-intercept \( \quad(x, y)=(\square \)-intercept \( \quad(x, y)=(\square) \)

Unit 3 AsSessment Path of Lava Lava coming from the eruption of a volcano follows a parabolic path. The height \( h \) in feet of a piece of lava \( t \) seconds after it is ejected from the volcano is given by \( h(t)=-t^{2}+16 t+936 \).

Consider the following. \[ g(x)=\frac{x^{2}-2 x-35}{x^{2}-36} \] (a) State the domain of the function. (b) Identify all intercepts. (c) Find any vertical and horizontal asymptotes or holes. (d) Plot additional solution points as needed to sketch the graph of the rational function.