1. Given the function f, prove that f is one-to-one using the definition of a one-to-one function.
f(x) = 4 - 2x
2. Graph the equation by substituting and plotting points. Then reflect the graph across the line y = x to obtain the graph of its inverse.
y = x² + 1
3. For the function f, use composition of functions to show that f^(-1) is as given.
f(x) = (x+5)/(4 ) f^(-1) (x) = 4x - 5
4. Find each of the following, to four decimal places, using a calculator.
( 1/e^3 )2
log 0.082
ln 0
5. Sketch the graph and use the graph to determine whether the function is one-to-one.
b) If the function is one-to-one, find a formula for the inverse.
f(x) = 7 - x
6. Convert to an exponential equation.
t = log_4 7
ln 0.38 = - 0.9676
log_t Q = k
7. Find each of the following. Do not use a calculator.
log_2 64
ln e
log 1
log 10^(8/5)
log_2 √2
log_64 4
8. Convert to a logarithmic equation.
e^(-1 )=0.3679
5^(-3)= 1/125
y= 5^x
9. Graph.
f(x) = ln x
10. The number of foreign nationals who came to the United States to marry an American using a "fiancée Visa" has grown exponentially in recent years. The total number of fiancée visas is given by the function
f(x) = 5728.98 ( 1.1214)^x ,
Where x is the number of years since 1990 (Source: US Department of Homeland Security).
Find the total number of Visas in 2006.
Find the total number of Visas in 2009.
11. The value of a stock is given by the function
V(t) = 58 ( 1 - e^(-1.1t) ) + 20 ,
Where V is the value of the stock after time t, in months. Find V (1), V (2), V (4), V (6), and V (12).
12. The bacteria Escherichia Coli are commonly found in the human intestines. Suppose that 3,000 of the bacteria are present at time t = 0. Then under certain conditions, t minutes later, the number of bacteria present is
N (t) = 3000 (2)^(t/20)
How many bacteria will be present after 10 minutes? 20 minutes? 30 minutes? 40 minutes? And 60 minutes?