Exponential Functions Notes

Exponential Functions

• Definitions and Key Concepts

  • Exponential Functions: Functions of the form $f(x) = ab^x$ where:
  • $a$ is a non-zero real number (the initial value).
  • $b$ is a positive real number, with $b <br />\neq 1$ (growth if b > 1, decay if 0 < b < 1).
  • Domain: All real numbers ((-,))
  • Range:
  • If a > 0, range is (0,)$.
  • If a < 0$, range is$(−,0)$.
  • Horizontal Asymptote: $y = 0$
  • y-Intercept: $(0, a)$
  • Growth vs. Decay:
  • Growth: If b > 1, values increase rapidly as $x$ increases.
  • Decay: If 0 < b < 1, values decrease as $x$ increases.

• Evaluation of Exponential Functions:

  • Problem Example:
  • Given $f(x) = 5(3)^{x+1}$, evaluate for different $x$ values.
  • Analyze other forms such as $g(x) = 5x^2 + 3$ and $h(x) = 2(1/2)^{x+1}$.

• Finding Equations of Exponential Functions:

  • Steps to determine the formula based on given points:
  • Example: Points (0, 6) and (3, 750)
    • Plugging point (0, 6):
    • 6 = ab^0
      ightarrow a = 6
    • Plugging point (3, 750):
    • 750 = 6b^3
      ightarrow b^3 = 125
      ightarrow b = 5
    • Resultant function: $f(x) = 6(5^x)$

• Compound Interest Formula:

  • Given by: $A = P(1 + \frac{r}{n})^{nt}$
  • Where:
  • $A$ = amount of money accumulated after n years, including interest.
  • $P$ = principal amount (the initial amount of money).
  • $r$ = annual interest rate (decimal).
  • $n$ = number of times interest applied per time period.
  • $t$ = number of time periods the money is invested or borrowed.
  • Example problems:
  • Invest $3,000 at 3% interest compounded quarterly for 10 years, find the account worth.
  • How much to invest to grow to $40,000 over 18 years at 6% compounded semi-annually?

• The Number e:

  • $e ext{ (approx. 2.718282)}$ is crucial for continuous growth/decay calculations.
  • Origin of $e$ linked to the limit of compound interest as n approaches infinity: $e = (1 + \frac{1}{n})^n$.

• Continuous Growth / Decay Formula:

  • $A(t) = ae^{rt}$
  • Where:
    • $a$ = initial value
    • $r$ = growth rate (if r > 0) or decay rate (if r < 0)
    • $t$ = time
  • In business context, use $A(t) = Pe^{rt}$ for continuous compounding:
  • $P$ = principal, $r$ = growth/interest rate, $t$ = time.
  • Example problem with Radon-222 decay at 17.3% continuous rate over 3 days starting from 100 mg.

• Characteristics of exponential graphs:

  • Graphs of $f(x) = 2^x$ increase steeply, show growth.
  • As $x$ approaches negative infinity, $f(x)$ approaches 0 (never touches the x-axis).
  • Key features: increasing function, y-intercept at (0, 1), horizontal asymptote at y=0.