For each function: $\\$ a) Determine whether it is one-to-one.$

$$b) If the function is one-to-one, find a formula for the inverse.$$

f(x)=(x+5)^3$

Find each of the following using a calculator. Round to four decimal places. $\\$$\log 0.082$

Interest in a College Trust Fund. Following the birth of her child, Sophia deposits $\$ 10,000$ in a college trust fund where interest is $3.9 \%$, compounded semiannually.

a) Find a function for the amount in the account after $t$ years.

b) Find the amount of money in the account at $t=0,4,8,10,18$, and 21 years.

Solve the exercise below: $\\$$2 \ln x-\ln 5=\ln (x+10)$

Given the function below:

a) Sketch the graph and determine whether the function is one-to-one.

b) If it is one-to-one, find a formula for the inverse.

$$f(x)=\frac{x}{2-x}$$

Fiancé(e) Visas. The number of foreign nationals who came to the United States to marry an American using a "fiancé(e) visa" has grown exponentially in recent years. The total number of fiancé(e) visas issued is given by the function

$$f(x)=5728.98(1.1214)^x,$$

where $x$ is the number of years since 1990 (Source: U.S. Department of Homeland Security). Find the total number of fiancé(e) visas issued in 2006 and in 2009.

Use the compound-interest formula to find the account balance A with the given conditions: $\\$$P=$principal,$r=$interest rate,$n=$number of compounding periods per year,$t=$time, in years,$A=$account balance.

Solve the exponential equation algebraically. Then check using a graphing calculator. $\\$$2^x=40$

The number of U.S. troops diagnosed as overweight has more than doubled since 1998. The exponential function

$$W(x)=23,672.16(1.112)^x,$$

where $x$ is the number of years since 1998, which can be used to estimate the number of overweight active troops. Find the number of overweight troops in 2000, 2008, and in 2013.

The central processing unit (CPU) power in computers has increased significantly over the years. The CPU power in Macintosh computers has grown exponentially from $8 \mathrm{MHz}$ in 1984 to $3400 \mathrm{MHz}$ in 2013 (Source: Apple). The exponential function

$$M(t)=7.91477(1.26698)^t,$$

where $t$ is the number of years after 1984, can be used to estimate the CPU power in a Macintosh computer in a given year. Find the CPU power of a Macintosh Performa $5320$ $\mathrm{CD}$ in 1995 and of an iMac G6 in 2009. Round to the nearest one $\mathrm{MHz}$.

Make a hand-drawn graph of each of the following. Then check your work using a graphing calculator. $\\$$y=\log _4 x$

Express as a sum of logarithms. $\\$$\log 0.2 x$

Find an equation of the inverse relation. $\\$$2 x^2+5 y^2=4$

Use the compound-interest formula to find the account balance A with the given conditions: $\\$$P=$principal,$r=$interest rate,$n=$number of compounding periods per year,$t=$time, in years,$A=$account balance.