College Algebra Study Notes: Compound Interest and Inverse Functions

Compound Interest

Understanding Compound Interest

Compound interest is a method of calculating interest where the interest earned is added to the principal, so that from that moment on, the interest that has been added also earns interest. This is a common way banks calculate interest on savings accounts.

• Principal (P): The initial amount of money invested or loaned.

• Annual Interest Rate (r): The percentage rate at which interest is earned annually.

• Number of Compounding Periods per Year (n): How many times per year the interest is compounded (e.g., monthly means n = 12).

• Time (t): The number of years the money is invested or borrowed.

The formula for compound interest is:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

• A: The amount of money accumulated after n years, including interest.

• P: The principal amount (the initial investment).

• r: The annual interest rate (in decimal form).

• n: The number of times that interest is compounded per year.

• t: The number of years the money is invested.

Example: Calculating Future Value

Problem: You invest $20,000 at a bank that compounds monthly at an annual interest rate of 3%.

• P = $20,000

• r = 0.03

• n = 12

• t = number of years

Formula:

$A = 20000 \left(1 + \frac{0.03}{12}\right)^{12t}$

This formula gives the total amount in the account after t years.

Example: Tripling Your Investment

Problem: How long will it take for your $20,000 investment to triple?

• Set $A = 3 \times 20000 = 60000$

• Use the compound interest formula and solve for t:

$60000 = 20000 \left(1 + \frac{0.03}{12}\right)^{12t}$

Divide both sides by 20000:

$3 = \left(1 + \frac{0.03}{12}\right)^{12t}$

Take the natural logarithm of both sides:

$\ln(3) = 12t \cdot \ln\left(1 + \frac{0.03}{12}\right)$

Solve for t:

$t = \frac{\ln(3)}{12 \cdot \ln\left(1 + \frac{0.03}{12}\right)}$

This equation gives the number of years required to triple the investment.

Key Points

• Compound interest grows faster than simple interest because interest is earned on previously accumulated interest.

• The more frequently interest is compounded, the greater the total amount accumulated.

Inverse Functions

Definition and Properties

An inverse function reverses the effect of the original function. If $f(x)$ maps $x$ to $y$, then the inverse function $f^{-1}(x)$ maps $y$ back to $x$. Not all functions have inverses; a function must be one-to-one (each output corresponds to exactly one input).

• Notation: The inverse of $f(x)$ is written as $f^{-1}(x)$.

• Finding the Inverse: To find the inverse, solve the equation $y = f(x)$ for $x$ in terms of $y$, then swap $x$ and $y$.

• Verification: A function and its inverse "undo" each other: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Example: Finding the Inverse

Given: $f(x) = x^2 - 1$, $x \geq 0$

• Let $y = x^2 - 1$

• Solve for $x$:

$y = x^2 - 1$

$y + 1 = x^2$

$x = \sqrt{y + 1}$ (since $x \geq 0$)

Swap $x$ and $y$ to write the inverse:

$g(x) = f^{-1}(x) = \sqrt{x + 1}$

Verification

• Check $f(g(x))$:

$f(g(x)) = (\sqrt{x + 1})^2 - 1 = x + 1 - 1 = x$

• Check $g(f(x))$:

$g(f(x)) = \sqrt{x^2 - 1 + 1} = \sqrt{x^2} = x$ (since $x \geq 0$)

Domain and Range

• Original function $f(x) = x^2 - 1$:

  • Domain: $x \geq 0$

  • Range: $y \geq -1$

• Inverse function $g(x) = \sqrt{x + 1}$:

  • Domain: $x \geq -1$

  • Range: $y \geq 0$

Key Points

• The domain of a function becomes the range of its inverse, and vice versa.

• Always check that the inverse function is valid for the specified domain and range.