Pre-Calculus I: Elementary Functions

Functions and Evaluations

Function Composition Evaluations

  • Evaluate the following compositions and operations:
    • (a) $f ext{ o } g(5)$
    • (b) $f ext{ o } f(0)$
    • (c) $g ext{ o } f(7)$
    • (d) $g^{-1}(3)$
    • (e) $g(5) + f(8)$
    • (f) $(4)(10)$

Graph Analysis

  • Graph of function $f(x)$ includes points:
    • $y = 9$ at $f(4)$
    • $y = 8$ at some points
    • $y = 7$ around $x = 3$
    • $y = 6$, $y = 5$, $y = 4$ are all continuously decreasing as $x$ increases (decreasing function)
  • Evaluate the following based on provided graphs:
    • (a) $f ext{ o } g(1)$ at coordinates regarding point intersection
    • (b) $g^{-1}(3)$ with restricted domain $x < -1$
    • (c) $g ext{ o } f(x) = -2$ using graphical point relationship

Exponential Functions

Writing Exponential Equations

  1. Given parameters:

    • a = 12, r = 7, n = 12
    • Write the equation:
      A(t)=a(1+r)nA(t) = a(1 + r)^n
      A(t)=12(1+7)12A(t) = 12(1 + 7)^{12}

  2. Given a = 300, continuously decreasing by 3.1:

    • Write the exponential equation:
      A(t)=300e−3.1tA(t) = 300 e^{-3.1t}

Function Models Based on Point Data

Constructing Functions

  1. Given points (3, 2) and (-1, 8) for function $g$:
    • (a) Linear function model
    • (b) Exponential function model
    • (c) Graph sketching each function on the same axis to illustrate the difference in growth rates

Inverse Functions

Finding Inverses

  1. Following five steps to find inverse function:
    • Given $f(x) = 5 + 4x$, rearrange to find inverse, verify by plugging back the values (explain each step thoroughly).

Verification of Inverses

  • Check if two functions are inverses:
    • Compute $f(g(x))$ and $g(f(x))$, ensuring both equal $x$.
    • Graphical check: Reflect original function over line $y = x$.
    • For given functions $f(x) = ax + b$ and $g(x) = c ext{s},$ verify inverses through algebra and reflection.

Compound Interest Calculations

Compounding Details

  1. Given account earns an APR of 3% compounded hourly:

    • (a) Identify $n$ and $r$:
      • $n = 365$ (hourly compounding)
      • $r = 0.03$
    • (b) Write balance equation after 2 years when balance = $318.56$:
      A(t) = P imes (1 + rac{r}{n})^{nt}
    • (c) Effective annual factor calculation
    • (d) Effective annual rate calculation
    • (e) Comparison: nominal vs effective

  2. Given point (5,800) on graph of function $Q(t)$ and $a = 150$:

    • (a) Solve for $k$ using equation set-up
    • (b) Continuous exponential equation formation based upon $Q(t)$
    • (c) State both effective annual factor and effective annual rate when needed.

Comparative Analysis of Investment Accounts

Analyses for Various Accounts

  1. Three accounts comparison scenario:
    • Account 1: 8.2% nominal rate, monthly compounded
    • Account 2: 8% nominal rate, daily compounded
    • Account 3: 7.5% compounded continuously
    • (a) Formula for each account, $B(t) = P(1 + rac{r}{n})^{nt}$ for non-continuous compounding; equivalent for continuous compounding.
    • (b) Calculate balance for each after 1 year
    • (c) Returns comparison over a 15-year span.
    • (d) Select the better investment based upon return and explain the basis of selection.

Population Growth Analysis

Continuous Growth of Bacteria

  1. Given bacteria population at a constant growth rate of 9.2% per day:
    • (a) Starting population: 1500, write growth representation
      P(t)=P0ektP(t) = P_0 e^{kt} where $k = 0.092$
    • (b) Determine population after 30 days
    • (c) Find growth factor $b$ from the equation
    • (d) Solve for $t$ if $P = 10355$ both algebraically and graphically.

Logarithmic Properties and Problems

Understanding Logarithms

  • Definition of logarithm includes:
    extlogb(x)=yextifby=xext{log}_b(x) = y ext{ if } b^y = x
  • Natural logarithm properties:
  1. $ ext{log}(ab) = ext{log}(a) + ext{log}(b)$
  2. $ ext{log}(a^t) = t ext{log}(a)$
  3. $ ext{log}(1) = 0,$ and $ ext{log}(b) = 1, ext{ln}(e) = 1$.

Logarithm Evaluation Tasks

  • Fill in values for the logarithmic statements:
    (a) $ ext{log}(36) = 2$, (b) $ ext{log}_3(27)$,
    and others as specified as part of evaluation exercises.

  • Solve for x in equations involving logarithms or exponents using logarithmic properties, consider:

    • (a) $ ext{log}(6x^3) = 8$
    • (b) $10 ext{%} = 3(1.5)$,
    • (c) $1.364 = e^k$ and other specified problems.

General Concepts in Pre-Calculus Functions

Review of Concepts Learned

  • Topics covered include:
    • Exponentials, definitions, variable impacts on graphs
    • Inverses, identifying and proving inverses, graphical reflections
    • Compound interest calculations, effects of different rates and their representations
    • Logarithmic concepts and properties applicable to solving various equations.

Function Evaluation on Standards

  • Evaluation/simplification of compositions of two or more functions as seen with:
    • Given $f(x)=3x+5$ and $g(x)=x-1$ evaluate $f(g(2))$, $f ext{o} f(-3)$,…
    • Offering thorough breakdown of each function evaluation and rationale for process followed.