Find the domain of the logarithmic function \( y=\log (3 x+7) \) analytically. You may wish to check your answer graphically. The domain is \( (\square, \infty) \). (Simplify your answer. Type an integer or fraction.)

Consider the following. \[ \begin{array}{l}g(x)=\frac{5 x}{x^{2}+4 x-5} \\ \text { (a) State the domain of the function. } \\ \text { all real numbers } x \text { except } x=5 \\ \text { all real numbers } x \text { except } x=1 \\ \text { all real numbers } x \text { except } x=-5 \\ \text { all real numbers } x \text { except } x=-5 \text { and } x=1 \\ \text { (b) Identify all intercepts. (If an answer does not exist, enter DNE.) } \\ x \text { except } x=0 \\ x \text {-intercept } \quad(x, y)=(\square) \\ y \text {-intercept } \quad(x, y)=(\square)\end{array} \]

Consider the following. \[ \begin{array}{l}g(x)=\frac{6(x+3)}{x(x-6)} \\ \text { (a) State the domain of the function. } \\ \text { all real numbers } x \text { except } x=3 \text { and } x=6 \\ \text { all real numbers } x \text { except } x=0 \text { and } x=6 \\ 0 \text { all real numbers } x \text { except } x=-3 \text { and } x=6 \\ 0 \text { all real numbers } x \text { except } x=0 \text { and } x=3 \\ \text { all real numbers } x \\ \text { (b) Identify all intercepts. (If an answer does not exist, enter DNE.) } \\ x \text {-intercept } \quad(x, y)=(\square)\end{array} \] \( y \)-intercept \( \quad(x, y)=(\square) \)

Consider the following. \[ f(x)=\frac{x^{2}+4}{x+1} \] (a) State the domain of the function. (b) Identify all intercepts. (c) Find any vertical and slant asymptotes. (d) Plot additional solution points as needed to sketch the graph of the rational function.

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. \( k=0.19 ; P(t)=550 e^{0.19 t} \)

The number of bacteria growing in an incubation culture increases with time according to \( B(x)=9,900(5)^{x} \), where \( x \) is time in days. Find the number of bacteria when \( x=0 \) and \( x=4 \). A. \( 9,900,30,937,500 \) B. \( 49,500,6,187,500 \) C. \( 9,900,6,187,500 \) D. \( 9,900,198,000 \)