Precalculus Study Notes: Functions, Factoring, Domains, and Trigonometric Equations

Functions and Their Properties

Piecewise Functions

Piecewise functions are defined by different expressions depending on the input value. They are useful for modeling situations where a rule changes based on the domain.

• Definition: A function defined by multiple sub-functions, each applying to a certain interval of the domain.

• Example: If $f(x) = 2x + 4$ for $x \geq 0$ and $f(x) = x + 2$ for $x < 0$, then:

$\begin{cases} f(x) = 2x + 4 & \text{if } x \geq 0 \\ f(x) = x + 2 & \text{if } x < 0 \end{cases}$

• Domain: The set of all $x$ values for which the function is defined.

Function Operations and Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and is a precursor to calculus.

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = 3x$:

$\begin{align*} f(x+h) &= 3(x+h) = 3x + 3h \\ f(x+h) - f(x) &= (3x + 3h) - 3x = 3h \\ \frac{f(x+h) - f(x)}{h} &= \frac{3h}{h} = 3 \end{align*}$

• Application: Used to find the slope of a function at a point.

Polynomial and Rational Functions

Factoring Polynomials

Factoring is the process of expressing a polynomial as a product of its factors. This is essential for solving equations and simplifying expressions.

• Common Methods: Factoring out the greatest common factor, grouping, and using formulas like difference of squares.

• Example: $x^2 + 5x + 6$

$(x+2)(x+3)$

• Difference of Squares: $a^2 - b^2 = (a-b)(a+b)$

• Example: $x^2 - 4 = (x-2)(x+2)$

Factoring by Substitution

Sometimes, substitution is used to simplify factoring, especially for higher-degree polynomials.

• Example: Let $u = x$, then factor $x^2 + 4x$ as $x(x+4)$.

Domain of Rational Functions

The domain of a rational function excludes values that make the denominator zero.

• Example: For $f(x) = \frac{x-4}{x^2-16}$, denominator is zero when $x^2-16=0$.

$(x-4)(x+4) = 0 \implies x = 4, x = -4$

• Domain: All real numbers except $x = 4$ and $x = -4$.

Trigonometric Functions and Identities

Trigonometric Values and Quadrants

Trigonometric functions have different signs depending on the quadrant.

• Cosine: Positive in Quadrants I and IV, negative in II and III.

• Sine: Positive in Quadrants I and II, negative in III and IV.

• Tangent: Positive in Quadrants I and III, negative in II and IV.

Trigonometric Identities

Identities are equations involving trigonometric functions that are true for all values in their domains.

• Sum and Difference Formulas:

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• Double Angle Formula: $\sin(2\theta) = 2\sin\theta\cos\theta$

• Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$

Solving Trigonometric Equations

Solving trigonometric equations often involves factoring, using identities, and considering the domain and range of the functions.

• Example: Solve $2\cos\theta + \cos\theta = 0$

$\cos\theta(2 + 1) = 0 \implies \cos\theta = 0$

• Example: Solve $2\sin\theta\cos\theta = \sin\theta$

$2\sin\theta\cos\theta - \sin\theta = 0 \\ \sin\theta(2\cos\theta - 1) = 0$

• Set each factor to zero: $\sin\theta = 0$ or $2\cos\theta - 1 = 0$

• Solutions: $\theta = 0, \pi, 2\pi$ or $\cos\theta = \frac{1}{2}$

Application: Finding Exact Values

Using sum and difference formulas to find exact values for trigonometric functions.

• Example: $\sin(150^\circ) = \sin(90^\circ + 60^\circ)$

$\sin(150^\circ) = \sin(90^\circ)\cos(60^\circ) + \cos(90^\circ)\sin(60^\circ) = 1 \times \frac{1}{2} + 0 \times \frac{\sqrt{3}}{2} = \frac{1}{2}$

Summary Table: Trigonometric Signs by Quadrant

Quadrant	Sine	Cosine	Tangent
I	+	+	+
II	+	-	-
III	-	-	+
IV	-	+	-

Additional info:

• Some steps and context were inferred from fragmented notes, such as domain restrictions and factoring methods.

• Trigonometric equation solutions were expanded for clarity.

• Examples were reconstructed for completeness.