College Algebra Test 3

Global Instructions: (100 points) Solve each problem and box in your final .

True/False questions. Enter ‘T’ for True and ‘F’ for False.

(a) The domain of a logarithm function is (−∞,∞).
(b) The range of the inverse of a function f is the same as the domain of f.
(c) If f is invertible, then f(f ^-1(w)) = w, for all w in the domain of f ^-1.
(d) To determine whether a function is one-to-one, we use the vertical line test .
(e) Suppose a > 0 and a ≠ 1. If a^u = a^v, then it fol lows that u = v.

In the questions that follow, fill in the blank.

(f) Let a > 0 and a ≠ 1. Then if and only if t = .
(g) The natural logarithm is defined to be the logarithm with a base of a = .
(h) The inverse to the function is .
(i) For all x, and for all x > 0, .
(j) Let f(x) = 3x + 1 and g(x) = 2x² − 1. Then .

Let and . Find the function (f o g) and find its domain. Simply !
Solution: We use standard methods :

	x ≠ 0
	substitution
	simplify, note x ≠ 2

  Indicate which of the following functions are one-to-one, hence invertible.

The functions depicted in graphs are one-to-one. (List the number of each graph that
is one -to-one, separating each with a comma.)

  Find the inverse function to .

Solution: We follow the procedure:

  Solve each of the following equations. (Box in your final answer.)

(a) Solution:	(c) Solution: If , then 2x = 52. We deduce that . (d) Solution: Using the properties of logarithms, we deduce that 4x ln e = 12, or that 4x = 12, since ln e = 1. Thus, .
(b) Solution:

  Find the domain of the function . Use interval notation.
Solution: There is a natural restraint that we can only take logarithms of positive numbers .

We require that x + 1 > 0 or x > −1. In interval notation, the domain is (−1,∞).

  Calculate each of the following exactly, using the various properties of logarithms and exponentials.

Solution:

Solution: Immediate!

Use the properties of logarithms to expand the expression

Solution: We use the properties of logarithms

Elementary. Solve each of the following equations for x using the various properties of exponentials
and logarithms. Leave your solutions in list form, for example, , boxed in, of course.

(a) Solution:

(b) Solution: Note that x > 0.

Advanced. Solve each of the following equations for x using the various properties of exponentials and
logarithms . Leave your solutions in list form, for example, , boxed in, of course.

(a) Solution: After noting that x > 0, we apply the properties of logs:

(b) 5 · 23x = 8 (round the answer to 3 decimal places) Solution:

Compound Interest : Recall the following formulas

Answer each of the following about compound interest.

(a) An amount of $100 is invested at a annual rate of 10%, compounded continuously. Find the value
of the investment after two years .

Solution: From the formula we have:

(b) Find the principal needed to get $10, 000 after two years if the principal is to be invested at 12%
compounded continuously.

Solution: From the formula we have:

The number N of bacteria present in a culture at time t (in hours) obeys the function

(a) (2 pts) What is the growth rate of the bacteria?

(b) (3 pts) At what time, t, will the number of bacteria reach a population size of 1700. (Express your
answer either as an exact algebraic expression , or as an approximation with 3 digits of accuracy .)

Solution: We set N(t) = 1700 and solve for t.