Functions and Their Properties: Precalculus Study Notes

Functions and Graphs

Definition of a Function

A function is a relation that assigns to each element in a set called the domain exactly one element in a set called the range. Functions are fundamental objects in precalculus, used to model relationships between quantities.

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually f(x) or y-values).

• Function notation: $f(x)$ denotes the value of the function f at x.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: The function $f(x) = x^2$ has domain $(- , )$ and range $[0, )$.

Types of Functions

Functions can be classified by their algebraic form and properties.

• Linear Function: $f(x) = mx + b$

• Quadratic Function: $f(x) = ax^2 + bx + c$

• Polynomial Function: $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$

• Rational Function: $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$

• Exponential Function: $f(x) = a^x$, $a > 0$, $a \neq 1$

• Logarithmic Function: $f(x) = \log_a x$, $a > 0$, $a \neq 1$

Example: $g(x) = \frac{1}{x-6}$ is a rational function with domain $x \neq 6$.

Evaluating Functions

To evaluate a function, substitute the input value into the function's formula.

• Given $f(x) = 2x + 3$, find $f(4)$:

• $f(4) = 2 \times 4 + 3 = 8 + 3 = 11$

Example: If $h(x) = x^2 - 5x + 6$, then $h(2) = 2^2 - 5 \times 2 + 6 = 4 - 10 + 6 = 0$.

Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the domain.

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases}$

• To evaluate, determine which interval the input belongs to and use the corresponding formula.

Graphing Functions

Graphing is a visual way to understand the behavior of functions.

• Plot points by evaluating the function at selected x-values.

• Identify key features: intercepts, maxima/minima, asymptotes.

• For rational functions, note vertical asymptotes where the denominator is zero.

Example: The graph of $f(x) = \frac{1}{x-6}$ has a vertical asymptote at $x=6$.

Operations on Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Addition: $(f+g)(x) = f(x) + g(x)$

• Subtraction: $(f-g)(x) = f(x) - g(x)$

• Multiplication: $(fg)(x) = f(x) \cdot g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

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

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

Properties of Functions

Functions may have special properties such as being even, odd, or periodic.

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain.

• Periodic Function: $f(x + p) = f(x)$ for all $x$ and some period $p$.

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Summary Table: Types of Functions and Their Key Properties

Type	General Form	Domain	Range	Key Features
Linear	$f(x) = mx + b$	$(-\infty, \infty)$	$(-\infty, \infty)$	Straight line, constant rate of change
Quadratic	$f(x) = ax^2 + bx + c$	$(-\infty, \infty)$	$[\text{min}, \infty)$ or $(-\infty, \text{max}]$	Parabola, vertex, axis of symmetry
Rational	$f(x) = \frac{P(x)}{Q(x)}$	$x$ such that $Q(x) \neq 0$	Depends on $P(x)$ and $Q(x)$	Vertical/horizontal asymptotes
Exponential	$f(x) = a^x$	$(-\infty, \infty)$	$(0, \infty)$	Rapid growth/decay
Logarithmic	$f(x) = \log_a x$	$(0, \infty)$	$(-\infty, \infty)$	Inverse of exponential

Additional info: Some content and examples have been inferred based on standard precalculus curriculum and the context of fragmented notes.