Overview
1. For each of the following functions, match the function to the graph:
- F(x) = cube root of (x − 4) + 2
- F(x) = e^(x − 2) + 3
- F(x) = (1/2)^x − 3
- F(x) = log(x + 1) − 2
5. F(x) = (1/3)√(x − 2) + 3
- Is there any value of x that would cause this function to be undefined?
6. F(x) = log(x − 5) + 1
- Is there any value of x that would cause this function to be undefined?
- If there are restrictions on the domain, explain those restrictions.
7. F(x) = e^(x + 4) + 7
- Is there any value of x that would cause this function to be undefined?
- If there are restrictions on the domain, explain those restrictions.
8A. 5 = log base 9 of (x² + 18)
- Part I: Convert this logarithmic equation to an exponential equation.
- Part II: Simplify and solve the equation to find the two solutions.
8B. 2 = log(3x + 4)
- Part I: Convert this logarithmic equation to an exponential equation.
- Part II: Simplify and solve the equation to find the solution.
8C. e · e^(x + 1) = 1
- Part I: Combine the exponential expressions to produce a single exponential expression.
- Part II: Convert the equation from Part I to a logarithmic equation.
- Part III: Solve the equation.
9. The nth term of a geometric sequence is: aₙ = 27(0.1)^(n − 1)
- Part I: Write the first 5 terms of this sequence.
- Part II: Use the formula for the sum of a geometric series to find the sum of the first 10 terms of this sequence.
10. A lake is stocked with 1,500 young trout. The number of the original trout alive after x years is given by: T(x) = 1500e^(−0.4x)
- Part I: Let 300 be the value of T(x). Rewrite the exponential equation to show this.
- Part II: Use natural logarithms to solve this equation for x.
11. A hot bowl of soup cools according to Newton’s Law of Cooling: T(t) = 68 + 144e^(−0.04t)
- Part I: What was the initial temperature (at time t = 0) of the soup?
- Part II: What is the temperature of the soup when t = 15 minutes?
- Part III: To the nearest whole minute, at what time t is the soup’s temperature 125°F?
12. The number of people in a town of 10,000 who have heard a rumor is given by: N(t) = 10000 / (5 + 1245e^(−0.97t))
- Part I: On Day 0 (t = 0), how many people knew the rumor?
- Part II: On Day 5, how many people have heard the rumor? Round to the nearest whole person.
- Part III: How long will it be until 1,000 people in the town have heard the rumor?