Functions and Their Graphs

Tags: #◇Computer Science #◇CSMATH1 #◇Conceptual

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◇Overview

Functions are fundamental mathematical objects that map an input to exactly one output. A function $f$ from a set $A$ to a set $B$ assigns to each element $x$ in $A$ exactly one element $y$ in $B$, denoted as $f(x) = y$.

The graph of a function $f$ is the set of all ordered pairs $(x, f(x))$ in the coordinate plane. It provides a visual representation of the function's behavior, showing how the output varies with the input.

Key concepts in studying functions include: Domain: The set of all possible input values (x-values) for which the function is defined. Range: The set of all possible output values (y-values) that the function produces. Intercepts: Points where the graph crosses the x-axis ($y=0$) and y-axis ($x=0$). Symmetry: Identifying if a function is even ($f(-x) = f(x)$), odd ($f(-x) = -f(x)$), or neither. Asymptotes: Lines that the graph approaches as $x$ or $y$ tends to infinity.

Understanding these properties is crucial for accurately sketching and analyzing the graphs of various types of functions (polynomial, rational, exponential, logarithmic, trigonometric, etc.).

◇Important Formulas

Function Notation: $y = f(x)$ Vertical Line Test: A curve in the Cartesian plane is the graph of a function if and only if no vertical line intersects the curve more than once. Even Function: $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the y-axis). Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain (symmetric about the origin).

◇Worked Examples

Example 1: Evaluating a Function
Given $f(x) = 2x^2 - 3x + 1$, evaluate $f(2)$.
Solution:
Substitute $x = 2$ into the function:
$f(2) = 2(2)^2 - 3(2) + 1$
$f(2) = 2(4) - 6 + 1$
$f(2) = 8 - 6 + 1$
$f(2) = 3$

Example 2: Finding the Domain
Find the domain of $g(x) = \frac{1}{x - 3}$.
Solution:
The function is undefined when the denominator is zero.
$x - 3 = 0 \implies x = 3$
Therefore, the domain is all real numbers except $x = 3$. In interval notation: $(-\infty, 3) \cup (3, \infty)$.

◇References & Navigation

• Next Topic: ◇Introduction to Limits
• Related Notes: ◇Curve Sketching Examples, ◇Behavior of Functions and Their Graphs