The radioactive substance cesium-137 has a half-life of 30 years. The amount \( A(t) \) (in grams) of a sample of cesium-137 remaining after \( t \) years is given by the following exponential function. \[ A(t)=621\left(\frac{1}{2}\right)^{\frac{t}{30}} \] Find the initial amount in the sample and the amount remaining after 80 years. Round your answers to the nearest gram as necessary.

Question 3 (4 points) What is the range of the inverse function of the function \( f(x)=3-\sqrt{x+1} \) ? a) \( (-\infty,-1] \) b) \( [-1, \infty) \) c) \( [3, \infty) \) d) \( (-\infty, 3] \)

A small lake is stocked with a certain species of fish. The fish population is modeled by the function \( P=\frac{14}{1+4 e^{-0.9 t}} \) where \( P \) is the number of fish in thousands and \( t \) is measured in years since the lake was stocked. (a) Find the fish population after 3 years. (Round your answer to the nearest whole fish.) fish (b) After how many years will the fish population reach 7000 fish? (Round your answer to two decimal places.)

What is the range of the inverse function of the function \( f(x)=3-\sqrt{x+1} \) ? a) \( (-\infty,-1] \) b) \( [-1, \infty) \) c) \( [3, \infty) \) d) \( (-\infty, 3] \)

(a) Use the vertex formula to find the vertex. \( \begin{array}{ll}\text { (b) Find the intervals where } f \text { is increasing and where } f \text { is } \\ \text { decreasing. } \\ \begin{array}{ll}\text { 47. } f(x)=6-x^{2} & \text { 48. } f(x)=2 x^{2}-2 x+1 \\ \text { 49. } f(x)=x^{2}-6 x & \text { 50. } f(x)=-2 x^{2}+4 x+ \\ \text { 51. } f(x)=2 x^{2}-4 x+1 & \text { 52. } f(x)=-3 x^{2}+x-\end{array}\end{array} \).

Use technology to find points and then graph the function \( y=\sqrt{x+3}-7 \) following the instructions below. Plot at least four points with integer coordinates that fit on the axes below. Click a point to delete it.