Review of Functions, Trigonometry, Inverses, and Logarithms

Function Definitions and Properties

• Function Definition: A rule of correspondence that assigns to each element $x$ in domain set $A$ exactly one element $y$ in range set $B$.

• Variables: $x$ is the independent variable; $y = f(x)$ is the dependent variable.

• Domain Restrictions ("Illegals"):

  • Division by zero is undefined in rational functions (denominator cannot be $0$).

  • Taking an even root of a negative number is undefined over real numbers.

  • The Natural Domain is the largest set of real numbers for which the function formula is valid.

• Four Representations: Functions can be represented verbally (words), numerically (tables), visually (graphs), or algebraically (formulas).

• Vertical Line Test: A curve in the $xy$-plane is the graph of a function of $x$ if and only if no vertical line intersects the curve more than once.

• Zeros and Intercepts:

  • Zeros are found by solving $f(x) = 0$ for $x$.

  • The $y$-intercept is located at $(0, f(0))$.

• Intervals of Increase and Decrease:

  • Strictly Increasing: $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ on an interval.

  • Strictly Decreasing: $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ on an interval.

• Symmetry Properties:

  • Even Function: Satisfies $f(-x) = f(x)$; graph is symmetric with respect to the $y$-axis.

  • Odd Function: Satisfies $f(-x) = -f(x)$; graph is symmetric about the origin ($180^\text{o}$ rotation).

Function Combination and Composition

• Algebraic Combinations:

  • Sum: $(f+g)(x) = f(x) + g(x)$, Domain: $A \cap B$

  • Difference: $(f-g)(x) = f(x) - g(x)$, Domain: $A \cap B$

  • Product: $(fg)(x) = f(x)g(x)$, Domain: $A \cap B$

  • Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, Domain: $\{x \in A \cap B \mid g(x) \neq 0\}$

• Composite Functions:

  • Formed by substitution: $(g \circ f)(x) = g(f(x))$.

  • The domain of $(f \circ g)(x)$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

  • In general, $(g \circ f) \neq (f \circ g)$.

Basic Classes of Functions

• Linear Functions: $y = f(x) = mx + b$

  • Slope $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$ defines the constant rate of change and line direction.

  • Point-slope form: $y - y_1 = m(x - x_1)$.

  • Standard form: $ax + by = c$ (allows vertical lines $x = k$).

• Polynomial Functions: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a non-negative integer degree. Domain is $(-\infty, \infty)$.

  • Quadratic function ($n=2$): $P(x) = ax^2 + bx + c$. Roots determined by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

  • Discriminant $b^2 - 4ac > 0$ gives $2$ real solutions; $b^2 - 4ac = 0$ gives $1$ real solution; $b^2 - 4ac < 0$ gives no real solutions.

• Power Functions: $f(x) = a x^b$

  • If $b$ is a positive integer, even powers are even functions (parabola-like $y = x^2$) and odd powers are odd functions (similar to $y = x^3$).

  • Root functions: $f(x) = x^{1/n} = \sqrt[n]{x}$.

  • Reciprocal function: $f(x) = x^{-1} = \frac{1}{x}$, domain $\{x \mid x \neq 0\}$.

• Algebraic vs. Transcendental:

  • Algebraic: Functions constructed using addition, subtraction, multiplication, division, rational powers, and roots (e.g., rational functions $f(x) = \frac{P(x)}{Q(x)}$).

  • Transcendental: Functions that transcend algebra, such as trigonometric, exponential, and logarithmic functions.

• Graph Transformations (for positive constant $c$):

  • Vertical shift: $y = f(x) \pm c$ (shifted up or down by $c$ units).

  • Horizontal shift: $y = f(x - c)$ (shifted right by $c$ units); $y = f(x + c)$ (shifted left by $c$ units).

  • Vertical stretch: $y = c f(x)$ for c > 1$.\n * Reflection: y = -f(x)$(reflected across the$x$-axis);$y = f(-x)$(reflected across the$y-axis).\n\n# Trigonometric Functions\n\n* **Angle Measures**:\n * Degrees measure geometric angles; Radians measure arc length on a unit circle (r = 1).\n * Conversion relation: \pi\,\text{rad} = 180^\text{o} \implies 1\,\text{rad} = \frac{180^\text{o}}{\pi} \approx 57.3^\text{o}$and$1^\text{o} = \frac{\pi}{180}\,\text{rad} \approx 0.017\,\text{rad}.\n* **Unit Circle Definitions**:\n * For point P(x,y)$on the unit circle at angle$t$:$\cos(t) = x$and$\sin(t) = y.\n * \tan(t) = \frac{y}{x}$,$\csc(t) = \frac{1}{y}$,$\sec(t) = \frac{1}{x}$,$\cot(t) = \frac{x}{y}.\n * Domain of \sin(t)$and$\cos(t)$is$(-\infty, \infty)$; Range is$[-1, 1].\n\n\n\n* **Parity and Periodicity**:\n * Even functions: \cos(-t) = \cos(t)$,$\sec(-t) = \sec(t).\n * Odd functions: \sin(-t) = -\sin(t)$,$\tan(-t) = -\tan(t)$,$\csc(-t) = -\csc(t)$,$\cot(-t) = -\cot(t).\n * Period of \sin, \cos, \csc, \sec$is$2\pi$. Period of$\tan, \cot$is$\pi.\n* **Fundamental Identities**:\n * Pythagorean: \sin^2(t) + \cos^2(t) = 1$,$1 + \tan^2(t) = \sec^2(t)$,$1 + \cot^2(t) = \csc^2(t)\n * Addition: \sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)$,$\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)\n * Double-Angle: \sin(2x) = 2\sin(x)\cos(x)$,$\cos(2x) = \cos^2(x) - \sin^2(x)\n * Half-Angle: \sin^2(x) = \frac{1 - \cos(2x)}{2}$,$\cos^2(x) = \frac{1 + \cos(2x)}{2}\n* **Sinusoidal Transformations**: f(x) = A \sin(B(x - \alpha)) + C\n * Amplitude = |A|\n * Period = \frac{2\pi}{|B|}\n * Phase Shift (horizontal shift) = \alpha\n * Vertical Shift = C\n\n# Inverse Functions\n\n* **One-to-One Functions**: A function f$is one-to-one if$f(x_1) eq f(x_2)$whenever$x_1 eq x_2.\n* **Horizontal Line Test**: A function is one-to-one if and only if no horizontal line intersects its graph more than once.\n* **Definition & Domain/Range**:\n * The inverse function f^{-1}$reverses$f$:$f^{-1}(b) = a \iff f(a) = b.\n * If f$has domain$D$and range$R$, then$f^{-1}$has domain$R$and range$D.\n* **Cancellation Equations**: f^{-1}(f(x)) = x$for all$x \in D$, and$f(f^{-1}(y)) = y$for all$y \in R.\n* **Notation Caution**: f^{-1}(x) eq \frac{1}{f(x)}$. The reciprocal is written as$[f(x)]^{-1}.\n* **Steps to Find Inverse**:\n 1. Set y = f(x).\n 2. Solve algebraically for x$in terms of$y.\n 3. Interchange x$and$y$to write$y = f^{-1}(x).\n* **Graph Reflection**: The graph of f^{-1}$is obtained by reflecting the graph of$f$across the line$y = x.\n\n\n\n* **Restricted Domain**: Non-one-to-one functions (e.g., f(x) = x^2$) require restricting the domain (e.g.,$x \ge 0$) to define an inverse function ($g^{-1}(x) = \sqrt{x}).\n\n# Exponential and Logarithmic Functions\n\n* **Exponential Functions**: f(x) = a^x$($a > 0, a eq 1)\n * Domain is (-\infty, \infty)$, Range is$(0, \infty)$, asymptotic to the$x-axis.\n * Base e = 2.817..$yields the **Natural Exponential Function**$f(x) = e^x$.

• Laws of Exponents:

  • $a^x a^y = a^{x+y}$

  • $\frac{a^x}{a^y} = a^{x-y}$

  • $(a^x)^y = a^{xy}$

  • $(ab)^x = a^x b^x$

  • $\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$

• Logarithmic Functions: The inverse of exponential functions: x = \log_a(y) \iff y = a^x$.\n * Domain is (0, \infty)$, Range is$(-\infty, \infty).\n* **Properties of Logarithms**:\n * \log_a(1) = 0\n * \log_a(bc) = \log_a(b) + \log_a(c) (Product Property)\n * \log_a\left(\frac{b}{c}\right) = \log_a(b) - \log_a(c) (Quotient Property)\n * \log_a(b^x) = x \log_a(b) (Power Property)\n * \log_a(a^x) = x$and$a^{\log_a(x)} = x$for$x > 0\n* **Change of Base Formula**: \log_a(x) = \frac{\ln(x)}{\ln(a)}$$