3. En el presente problema trabajamos con restricciones de funciones. Sea la función \[ f: \mathbb{R} \rightarrow \mathbb{R}, \quad f(x)=10+20 \cos \left(\frac{2 \pi}{32}(x-7)\right) \text {. (2) } \] Sea la función restricción \( g=\left.f\right|_{[39,71]} \). La imagen de la función \( g \) es rango( \( g)=[A, B] \). (a) Encuentre el valor de \( A \). (b) Encuentre el valor de \( B \). La imagen inversa de 10 satisface \( g^{-1}(10)=\{C, D\} \) con \( C<D \). (c) Encuentre el valor de \( C \). (d) Encuentre el valor de \( D \). Considere la función restricción \( h=\left.f\right|_{[39, E]} \), donde \( E \) es una con- stante a ser determinada. (e) Encuentre máximo valor posible de \( E \in \mathbb{R} \) de manera que \( h \) sea inyectiva.

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 ?

The function \( f(t)=60,000(2)^{\frac{1}{40}} \) gives the number of bacteria in a population \( t \) minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?

What is the direction of a vector with a horizontal component of 2 and a vertical component of -4 ?

Find an equation for the hyperbola described. Graph the equation. Center at \( (-1,-7) ; \) focus at \( (-1,-10) ; \) vertex at \( (-1,-9) \) Write an equation for the hyperbola. \( \square-\square=1 \) (Type exact answers for each term, using fractions as needed.)

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.)