Suppose that the functions \( f \) and \( g \) are defined as follows. \[ \begin{array}{l}f(x)=\sqrt{5 x+3} \\ g(x)=3 x^{2}+1\end{array} \] Find \( f+g \) and \( f \cdot g \). Then, give their domains using interval notation.

The number of U.S. travelers to other countries during the period from 1990 through 2009 can be modeled by the polynomial function \( P(x)=-0.00530 x^{3}+0.1366 x^{2}+1.250 x+46.49 \) where \( x=0 \) represents \( 1990, x=1 \) represents 1991 , and so on and \( P(x) \) is in millions. Use this function to approximate the number of \( U \).S. travelers to other countries in each given year. (a) 1990 (b) 2000 (c) 2009 (a) In 1990 , the number of U.S. travelers to other countries was approximately \( \square \) million. (Simplify your answer. Type an integer or decimal rounded to the nearest tenth as needed.) (b) In 2000, the number of U.S. travelers to other countries was approximately \( \square \) million. (Simplify your answer. Type an integer or decimal rounded to the nearest tenth as needed.) (c) In 2009, the number of U.S. travelers to other countries was approximately \( \square \) million. (Simplify your answer. Type an integer or decimal rounded to the nearest tenth as needed.)

Determine the implied domain of the following function. Express your answer in interval notation. \[ f(x)=\sqrt{x}-9 \] Answer

The size of a certain insect population is given by \( \mathrm{P}(\mathrm{t})=400 e^{.03 \mathrm{t}} \), where t is measured in days. At what time will the population equal 1600 ?

Use the remainder theorem to find the remainder when \( f(x) \) is divided by \( x-1 \). Then use the factor theorem to determine whether \( x-1 \) is a factor of \( f(x) \). \( f(x)=2 x^{4}+7 x-3 \) The remainder is

Use the remainder theorem to find the remainder when \( f(x) \) is divided by \( x-1 \). Then use the factor theorem to determine whether \( x-1 \) is a factor of \( f(x) \). \( f(x)=3 x^{3}-2 x^{2}+7 x-2 \) The remainder is \( \square \).