Precalculus Study Guide: Power, Rational, Periodic, and Trigonometric Functions

Section 7.1: Combining Functions

Overview of Function Operations

Combining functions is a foundational concept in precalculus, involving operations such as addition, subtraction, multiplication, division, and composition of functions. Understanding how to combine functions allows for the construction of more complex mathematical models.

• Function Addition/Subtraction: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$

• Function Multiplication/Division: $(f \cdot g)(x) = f(x) \cdot g(x)$, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (where $g(x) \neq 0$)

• Function Composition: $(f \circ g)(x) = f(g(x))$

Example: If $f(x) = x^2$ and $g(x) = x+1$, then $(f+g)(x) = x^2 + x + 1$ and $(f \circ g)(x) = f(g(x)) = (x+1)^2$.

Section 7.2: Power Functions

Definition and Properties

Power functions are functions of the form $f(x) = ax^n$, where $a$ and $n$ are constants. They serve as a basis for understanding polynomial and rational functions.

• Even Power Functions: Functions like $f(x) = x^2$ are symmetric about the y-axis.

• Odd Power Functions: Functions like $f(x) = x^3$ are symmetric about the origin.

• End Behavior: The degree and sign of $a$ determine the end behavior of the graph.

Example: $f(x) = x^4$ is an even power function; its graph opens upwards and is symmetric about the y-axis.

Section 8.1-2: Polynomial Functions

Characteristics and Graphs

Polynomial functions are sums of power functions with non-negative integer exponents. Their graphs are smooth and continuous, and their degree determines the number of turning points and end behavior.

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0$

• Degree: The highest exponent $n$ in the polynomial.

• Zeros: The values of $x$ where $f(x) = 0$; these are the x-intercepts.

• End Behavior: Determined by the leading term $a_nx^n$.

Example: $f(x) = x^3 - 2x^2 + x - 5$ is a cubic polynomial with degree 3.

Section 8.3-4: Rational Functions

Definition and Asymptotic Behavior

Rational functions are quotients of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$. Their graphs can have vertical and horizontal asymptotes, and sometimes oblique asymptotes.

• Vertical Asymptotes: Occur where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$.

• Oblique Asymptotes: Occur when the degree of $P(x)$ is one more than the degree of $Q(x)$.

Example: $f(x) = \frac{2x^2 + 1}{x - 3}$ has a vertical asymptote at $x = 3$.

Type of Asymptote	Condition	Equation
Vertical	$Q(x) = 0$	$x = a$
Horizontal	Degree $P(x) <$ Degree $Q(x)$	$y = 0$
Horizontal	Degree $P(x) =$ Degree $Q(x)$	$y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$
Oblique	Degree $P(x) =$ Degree $Q(x) + 1$	Quotient from polynomial division

Section 1.1: Periodic Functions

Definition and Properties

Periodic functions repeat their values in regular intervals. The most common examples are trigonometric functions such as sine and cosine.

• Period: The smallest positive value $T$ such that $f(x + T) = f(x)$ for all $x$.

• Amplitude: The maximum absolute value of the function from its mean position.

• Frequency: The number of cycles per unit interval.

Example: $f(x) = \sin(x)$ has period $2\pi$ and amplitude $1$.

Section 1.2-1.5: Trigonometric Functions

Basic Properties and Applications

Trigonometric functions relate angles to ratios of sides in right triangles and are fundamental in modeling periodic phenomena.

• Definitions: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Unit Circle: The coordinates $(\cos(\theta), \sin(\theta))$ represent the terminal point of angle $\theta$.

• Special Angles: $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ and their radian equivalents.

Example: $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(60^\circ) = \frac{1}{2}$

Section 1.4: Inverse Trigonometric Functions

Solving for Angles

Inverse trigonometric functions allow us to determine angles from known ratios. They are denoted as $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$.

• Principal Values: Each inverse function has a restricted range to ensure it is a function.

• Multiple Solutions: On the unit circle, many angles can have the same sine, cosine, or tangent value.

• Applications: Used to solve triangles and model periodic phenomena.

Example: $\sin^{-1}(\frac{1}{2}) = 30^\circ$ or $\frac{\pi}{6}$

Section 2.1-2.2: Transformations on Sine and Cosine

Graphical Transformations

Transformations alter the amplitude, period, phase shift, and vertical shift of sine and cosine functions. The general form is $y = a \sin(bx + c) + d$.

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Phase Shift: $-\frac{c}{b}$

• Vertical Shift: $d$

Example: $y = 3 \sin(2x - \pi) + 1$ has amplitude $3$, period$\pi$, phase shift$\frac{\pi}{2}$, and vertical shift $1$.

Additional info: Some sections referenced external web resources for further examples and practice, but the main concepts are covered above for self-contained study.