Mathematics: Functions, Asymptotes, and Investment Calculations

Key Concepts in Functions and Asymptotes

• Domain & Range:

  • Domain: Set of possible input values for a function.

  • Range: Set of possible output values.

• Horizontal Asymptote:

  • Indicates the behavior of a function as the input approaches infinity.

  • General forms:

  • If $y = k$, then the horizontal asymptote is at $y = k$.

• Functions Notation:

  • Example: $f(x) = a x$ represents a linear function with constant $a$.

• Investments and Compounding:

  • Principal amount ($P$) and amount after $n$ compounding periods can be calculated using:

  • $A = P(1 + r/n)^{nt}$

  • Where $A$ is the amount, $r$ is the interest rate, $n$ is the number of compounding periods, and $t$ is time.

• Key Values:

  • Example calculations leading to various amounts (ex: future values) depending on compounding periods and interest rates.

  • Notable amounts:

  • $A = 13,826.47$

  • Future values will vary based on principal and rates used.

• Equations and Graphing:

  • Important to identify equation types and their asymptotic behavior when graphing functions.

  • Understand relation between changes in domain and range through transformations of functions.