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 $$ \sqrt{4-x^2}$ $$

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.

Express as a single logarithm and, if possible, simplify.

$$\ln \left(x^3-8\right)-\ln \left(x^2+2 x+4\right)+\ln (x+2)$$

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 $\mathrm{MHz}$.

Solve the exercise below: $\\$$e^x-2=-e^{-x}$

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.

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.

Given that $\log _b 2 \approx 0.693, \log _b 3 \approx 1.099$, and $\log _b 5 \approx 1.609$, find each of the following, if possible. Round the answer to the nearest thousandth.

$$\log _b \frac{5}{3}$$

In May 2010, a 1932 painting, Nude, Green Leaves, and Bust, by Pablo Picasso, sold at a New York City art auction for 106.5 dollars million to an anonymous buyer. This is a record price for any work of art sold at auction. The painting had belonged to the estate of Sydney and Francis Brody, who bought it for 17,000 dollars in 1952 from a New York art dealer, who had acquired it from Picasso in 1936.

Assuming that the value $A_0$ of the painting has grown exponentially:

a) Find the value of $k$, and determine the exponential growth function, assuming $A_0=17,000$ and $t$ is the number of years since 1952 .

b) Estimate the value of the painting in 2020.

c) What is the doubling time for the value of the painting?

d) After how long will the value of the painting be 240 million dollars, assuming there is no change in the growth rate?

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

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.

The number of students from India enrolled in U.S. colleges and universities has grown exponentially from about 9000 in 1980 to about 95,000 in 2008. The exponential function

$$I(t)=9000(1.0878)^t,$$

where $t$ is the number of years since 1980, which can be used to estimate the number of students from India enrolled in U.S. colleges and universities. Find the number of students from India enrolled in 1985, 1999, 2006, and in 2012.