Algebra II: Functions, Exponentials, and Logarithms Study Guide

Algebra of Functions and Composition

• Definitions of Operations on Functions: Let $f$ and $g$ be functions. New functions derived from $f$ and $g$ are defined as follows:     * Sum: $(f + g)(x) = f(x) + g(x)$     * Difference: $(f - g)(x) = f(x) - g(x)$     * Product: $(f \cdot g)(x) = f(x) \cdot g(x)$     * Quotient: $\frac{f}{g}(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$

• Example 1: Given $f(x) = x + 2$ and $g(x) = 3x + 5$, find:     * a) $(f + g)(x) = (x + 2) + (3x + 5) = 4x + 7$     * b) $(f - g)(x) = (x + 2) - (3x + 5) = -2x - 3$     * c) $(f \cdot g)(x) = (x + 2)(3x + 5) = 3x^2 + 11x + 10$     * d) $\frac{f}{g}(x) = \frac{x + 2}{3x + 5}$

• Composition Notation: The notation $(g \circ f)(x)$ means "$g$ composed with $f$" and is written as $g(f(x))$. This is often read as "$g$ of $f$ of $x$."

• Definition of Composition of Functions: The composition of functions $f$ and $g$ is defined as $(f \circ g)(x) = f(g(x))$.

• Example 2: If $f(x) = x^2 + 1$ and $g(x) = 3x - 5$, find:     * a) $(f \circ g)(x) = f(3x - 5) = (3x - 5)^2 + 1$     * b) $(f \circ g)(4) = f(g(4)) = f(3(4) - 5) = f(7) = 7^2 + 1 = 50$

• Example 3: If $f(x) = x + 5$ and $g(x) = x^2 - 2x$, find:     * a) $(f \circ g)(x) = (x^2 - 2x) + 5 = x^2 - 2x + 5$     * b) $(g \circ f)(x) = (x + 5)^2 - 2(x + 5) = (x^2 + 10x + 25) - 2x - 10 = x^2 + 8x + 15$

• Example 4: Decomposition of Functions. If $f(x) = 3x$, $g(x) = x - 4$, and $h(x) = |x|$, write each function as a composition using two of the given functions:     * a) $F(x) = |x - 4|$ is $(h \circ g)(x)$     * b) $G(x) = 3x - 4$ is $(g \circ f)(x)$

One-to-One Functions and Inverses

• One-to-One Function Definition: For a function to be one-to-one, each $x$-value (input) corresponds to exactly one $y$-value (output), and each $y$-value (output) corresponds to exactly one $x$-value (input).

• Example 5: Determining One-to-One Status:     * a) $f = \{(6, 2), (5, 4), (-1, 0), (7, 3)\}$: One-to-one (all unique outputs).     * b) $g = \{(3, 9), (-4, -2), (-3, 9), (0, 0)\}$: Not one-to-one (output $9$ repeats for inputs $3$ and $-3$).     * c) $h = \{(1, 1), (2, 2), (10, 10), (-5, -5)\}$: One-to-one.

• Horizontal Line Test: Every horizontal line must intersect the graph of a function at most once for the function to be one-to-one. This is used in conjunction with the vertical line test (which identifies if a relation is a function).

• Inverse Function Definition: The inverse of a one-to-one function $f$ is the function $f^{-1}$ consisting of all ordered pairs $(y, x)$ where $(x, y)$ was in the original function $f$.

• Example 7: Finding the Inverse of a Set:     * $f = \{(3, 4), (-2, 0), (2, 8), (6, 6)\}$     * $f^{-1} = \{(4, 3), (0, -2), (8, 2), (6, 6)\}$

• Procedure for Finding the Inverse Equation ($f(x)$):     * Step 1: Replace $f(x)$ with $y$.     * Step 2: Interchange $x$ and $y$.     * Step 3: Solve the equation for $y$.     * Step 4: Replace $y$ with the notation $f^{-1}(x)$.

• Example 8: Find the inverse of $f(x) = \frac{6 - x}{5}$     * $y = \frac{6 - x}{5}$     * $x = \frac{6 - y}{5}$     * $5x = 6 - y$     * $y = 6 - 5x$     * $f^{-1}(x) = 6 - 5x$

• Algebraic Verification of Inverses: Two functions $f$ and $f^{-1}$ are inverses if and only if:     * $(f^{-1} \circ f)(x) = x$     * $(f \circ f^{-1})(x) = x$

• Example 10: Prove $f(x) = 3x + 2$ and $f^{-1}(x) = \frac{x - 2}{3}$ are inverses.     * $(f^{-1} \circ f)(x) = \frac{(3x + 2) - 2}{3} = \frac{3x}{3} = x$     * $(f \circ f^{-1})(x) = 3(\frac{x - 2}{3}) + 2 = (x - 2) + 2 = x$

Exponential Functions

• Definition: An exponential function with base $b$ is denoted by $f(x) = b^x$, where $b > 0$, $b \neq 1$, and $x$ is any real number.     * The base cannot be $1$ because it would result in a constant function ($1^x = 1$).

• Behavior and Graphs:     * Each exponential graph has a horizontal asymptote at $y = 0$ (the $x$-axis).     * Transformations for $f(x) = 4^x$:         * $f(x) = 4^{x-1}$: Shift right $1$ unit.         * $f(x) = 4^x - 2$: Shift down $2$ units.         * $f(x) = -4^x + 5$: Reflect over $x$-axis and shift up $5$ units.         * $f(x) = 4^{-x}$: Reflect over $y$-axis.

• One-to-One Property of Exponents: For $b > 0$ and $b \neq 1$, $b^x = b^y$ if and only if $x = y$. This allows for solving simple exponential equations by finding common bases.

• Example 4 (Solving Equations):     * a) $3^x = 9 \rightarrow 3^x = 3^2 \rightarrow x = 2$     * b) $2^{2x-1} = 16 \rightarrow 2^{2x-1} = 2^4 \rightarrow 2x - 1 = 4 \rightarrow x = 2.5$     * c) $125^x = 25^{x-2} \rightarrow (5^3)^x = (5^2)^{x-2} \rightarrow 3x = 2x - 4 \rightarrow x = -4$     * d) $3^{x^2-3x} = 81 \rightarrow 3^{x^2-3x} = 3^4 \rightarrow x^2 - 3x - 4 = 0 \rightarrow (x-4)(x+1) = 0 \rightarrow x = 4, -1$

• The Natural Base $e$: Also known as Euler's number, discovered by Leonard Euler. $e \approx 2.718281828...$. The natural exponential function is $f(x) = e^x$.

Compound Interest Formulas

• Periodic Compounding: $A = P(1 + \frac{r}{n})^{nt}$     * $A$: Final balance     * $P$: Principal     * $r$: Annual interest rate (decimal)     * $n$: Number of compoundings per year     * $t$: Time in years

• Continuous Compounding: $A = Pe^{rt}$

• Example 5 (Investment Calculations): Investing $\$12,000$ at $3\%$ for $5$ years ($n$ values: Quarterly=4, Monthly=12, Daily=365):     * (A) Quarterly: $A = 12000(1 + \frac{0.03}{4})^{4 \times 5}$     * (B) Continuous: $A = 12000e^{0.03 \times 5}$

• Radioactive Decay Example: Percent $y$ of material after $t$ days: $y = 100(2.7)^{-0.1t}$. After $30$ days: $y = 100(2.7)^{-0.1(30)} = 100(2.7)^{-3}$.

Exponential Growth and Decay Applications

• General Concepts: Quantities growing or decaying by the same percentage at regular intervals represent exponential growth or decay.

• Population Growth Example: Town of Lacombe ($15,500$ in $2009$, increasing $10\%$ yearly).     * Prediction for $2029$ ($t = 20$): $P = 15500(1.10)^{20}$.

• Half-Life: The time required for half of a substance to decay.

• Example 3 (DDT Decay): Half-life of $15$ years. Initial amount of $400$ pounds. Final amount after $72$ years ($4.8$ half-lives):     * $A = 400(0.5)^{72/15}$

Logarithmic Functions

• Definition: $y = \log_b x$ means $x = b^y$. Requirements: $b > 0, b \neq 1, x > 0$.

• Conversion Examples:     * $\log_5 25 = 2 \leftrightarrow 5^2 = 25$     * $\log_6 (\frac{1}{6}) = -1 \leftrightarrow 6^{-1} = \frac{1}{6}$     * $4^3 = 64 \leftrightarrow \log_4 64 = 3$

• Evaluation and Solving:     * $\log_4 16 = 2$ (since $4^2=16$)     * $\log_9 3 = \frac{1}{2}$ (since $9^{1/2}=3$)     * Solve $\log_x 25 = 2 \rightarrow x^2 = 25 \rightarrow x = 5$

• Properties of Logarithms:     1. $\log_b 1 = 0$     2. $\log_b b = 1$     3. Inverse Property: $\log_b b^x = x$     4. One-to-One Property: If $\log_b x = \log_b y$, then $x = y$.

• Logs and Exponentials as Inverses: The graph of $y = \log_b x$ is the reflection of $y = b^x$ over the diagonal line $y = x$.     * Vertical asymptote for logs: $x = 0$.

Logarithmic Rules and Properties

• Product Property: $\log_b (xy) = \log_b x + \log_b y$

• Quotient Property: $\log_b (\frac{x}{y}) = \log_b x - \log_b y$

• Power Property: $\log_b x^r = r \log_b x$

• Condensing Logs (Single Logarithm):     * $2 \log_5 3 + 3 \log_5 2 = \log_5 3^2 + \log_5 2^3 = \log_5(9 \times 8) = \log_5 72$     * $\log_4 25 + \log_4 3 - \log_4 5 = \log_4 (\frac{25 \times 3}{5}) = \log_4 15$

• Expanding Logs:     * $\log_4 (\frac{a^2}{b^5}) = \log_4 a^2 - \log_4 b^5 = 2\log_4 a - 5\log_4 b$

Specialized Logarithms and Change of Base

• Common Logarithms: Base $10$. Written as $\log x$.     * Examples: $\log 100,000 = 5$, $\log 0.001 = -3$, $\log (\frac{1}{100}) = -2$.

• Richter Scale Formula: $R = \log(\frac{a}{T}) + B$ where $a$ is amplitude, $T$ is seconds between waves, and $B$ is adjustment factor.

• Natural Logarithms: Base $e$. Written as $\ln x$.     * Examples: $\ln e^3 = 3$, $\ln \sqrt[3]{e} = \frac{1}{3}$.

• Change of Base Formula: $\log_b a = \frac{\log_c a}{\log_c b}$. Usually used with common logs ($\log$) or natural logs ($\ln$) for calculator entry.     * $\log_5 3 = \frac{\ln 3}{\ln 5} \approx 0.6826$

Solving Equations and Advanced Problem Solving

• Logarithm Property of Equality: $\log_b a = \log_b c \leftrightarrow a = c$

• Solving Exponential Equations using Logs:     * $3^x = 7 \rightarrow \log 3^x = \log 7 \rightarrow x \log 3 = \log 7 \rightarrow x = \frac{\log 7}{\log 3} \approx 1.7712$

• Solving Logarithmic Equations:     * $\log_4(x - 2) = 2 \rightarrow x - 2 = 4^2 \rightarrow x = 18$     * $\log 2x + \log 2(x - 1) = 1 \rightarrow \log(2x(2x - 2)) = 1 \rightarrow 4x^2 - 4x = 10^1$

• Advanced Applications:     * Rabbit Population Growth: $y = y_0 e^{0.916t}$. For $y_0 = 60$ and $t = 3$: $y = 60 e^{0.916 \times 3}$.     * Investment Doubling Time: For $\$2000$ to double to $\$4000$ at $5\%$ compounded quarterly: $4000 = 2000(1 + \frac{0.05}{4})^{4t}$. Solve for $t$ using logarithms.