Question

part ii. logarithmic functions and their properties, continued

1. inverse relationships

a) find the inverse of the function (f(x) = 2^{x+1} - 4)
b) graph both the original function and its inverse on the same axes.

part iii. applications & critical thinking

1. financial growth investigation

suppose you invest $1000 in a savings account with a 5% annual interest rate, compounded continuously. the balance after (t) years is given by:
(a(t) = 1000e^{0.05t}).
a) calculate the account balance after 10 years.
b) find the time it takes for the balance to double (solve for (t)).
c) interpret the practical meaning of exponential growth in this situation.

Explanation:

Find the inverse function

Using the Logarithmic Functions and Function Transformations knowledge points

$$
LATEXBLOCK0
$$

Graph both functions

Using the Function Transformations knowledge point

$$
LATEXBLOCK1
$$

Calculate the account balance after 10 years

Substitute \(t = 10\) into the continuous compounding formula:

$$
LATEXBLOCK2
$$

Find the doubling time

Set the balance \(A(t) = 2000\) to find when the investment doubles:

$$
LATEXBLOCK3
$$

Interpret the practical meaning

Explain the real-world behavior of continuous interest:

• The account balance grows at a rate proportional to its current value.
• Interest is continuously earned and reinvested, leading to faster growth over time.

Answer:

Part II: Logarithmic Functions and Their Properties, continued

Question 2

a) The inverse function is:

$$f^{-1}(x) = \log_2(x + 4) - 1$$

b) The graphs of \(f(x) = 2^{x+1} - 4\) and \(f^{-1}(x) = \log_2(x+4) - 1\) are plotted on the same axes below. They are symmetric reflections across the line \(y = x\).

---

Part III: Applications & Critical Thinking

Question 1

a) The account balance after 10 years is:

$$\$1648.72$$

b) The time it takes for the balance to double is:

$$t = \frac{\ln(2)}{0.05} \approx 13.86\text{ years}$$

c) Practical interpretation:
Exponential growth means that the money in the savings account grows at a rate proportional to its current balance. As the balance increases, the amount of interest earned each instant also increases, causing the account value to grow faster and faster over time.