Pre Calculus Questions & Answers

Graph the exponential function. \[ f(x)=\frac{1}{4}(2)^{x} \] Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

Graph the exponential function. \[ f(x)=\left(\frac{1}{3}\right)^{-x} \] Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

Graph the exponential function. \[ f(x)=\left(\frac{1}{3}\right)^{x} \] Plot five points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function buttor

Sketch the graph of the exponential function \( g(x) = 3^{x-1} + 2 \), indicating its asymptote.

\& screenshot (4pts) Find the exact area of the region that lies inside the curve given (in polar form) by \( r=6 \sin (\theta) \) and outside the cardioid given by \( r=2+2 \sin (\theta) \).

\( P(t)=\frac{5(t-2)^{3}+30}{50+(t-2)^{2}} \) a) Calcular la población actual \( (t=0) \). b) Calcula la población en 10 años. c) Qué sucedio a medida que pasan los años

Country \( A \) has an exponential growth rate of \( 3.4 \% \) per year. The population is currently \( 4,406,000 \), and the land area of Country \( A \) is \( 24,000,000,000 \) square yards. Assuming this growth rate continues and is exponential, after how long will there be one person for every square yard of land?

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.

(3 screnshot (3pts) Find a Cartesian equation for the curve which is defined by the polar equation: \( r=6 \csc (\theta) \) where \( 0<\theta<\pi \).

6) \( \frac{36 \sqrt{5}+55 \sqrt{2}-97}{2 \sqrt{2}+\sqrt{5}-\sqrt{10}} \)