Introduction to Functions and Calculus Overview

Definition of a Function: A mapping or pairing between two sets such that each element in the first set is mapped or paired with only one element in the second set.
Mapping Example: * $3$ maps to $-7$ * $8$ maps to $15$ * $6$ maps to $0$ * $\text{star}$ maps to $\text{dollar sign}$
Terminology for Sets: * The First Set (Input): These elements are called the independent variables. Collectively, the entire set of input elements is referred to as the Domain. * The Second Set (Output): These elements are called the dependent variables. Collectively, this set is referred to as the Range.
Functional Notation: Functions are often described using equations, most traditionally represented as $y = f(x)$ . While $x$ and $y$ are standard tradition, variables can change depending on context (e.g., in economics, Price $P$ and Quantity $Q$ may be used).

Criterion for a Function: For every independent variable (input), there must be exactly one dependent variable (output).
Example 1: Polynomial Equation: In a function like $y = -2x^2 + 4x + 7$ , any value chosen for $x$ will result in only one possible value for $y$ . For instance: * If $x = 0$ , then $y = 7$ . * If $x = 1$ , then $y = 9$ .
Example 2: Linear Equation: To determine if $3x + 4y = 12$ is a function, solve for $y$ : * $4y = -3x + 12$ * $y = -\frac{3}{4}x + 3$ * Since substituting any value for $x$ yields exactly one $y$ value, this is a function.
Example 3: Circle Equation (Non-Function): Consider $x^2 + y^2 = 25$ . To determine if it is a function, solve for $y$ : * $y^2 = 25 - x^2$ * $y = \pm\sqrt{25 - x^2}$ * The Radicand Warning: The instructor warns specifically: "You cannot separate terms, things that are added or subtracted under a square root." Therefore, $\sqrt{25 - x^2}$ is NOT equal to $5 - x$ . * Testing Values: If $x = 0$ , then $y = \pm\sqrt{25}$ , which means $y = 5$ or $y = -5$ . Because one input ( $x = 0$ ) leads to two different outputs, this equation does NOT represent a function.

Names of Functions: Functions can be named using various letters such as $f(x)$ , $g(x)$ , $h(x)$ , or $m(x)$ .
Evaluating specific values for $f(x) = 3x + 2$ : * $f(3) = 3 \times 3 + 2 = 11$ (Meaning: when $x = 3$ , $y = 11$ ). * $f(-5) = 3 \times (-5) + 2 = -13$ . * $f(10) = 3 \times 10 + 2 = 32$ .
Evaluating Expressions: The input does not have to be a single number: * $f(abc) = 3 \times (abc) + 2 = 3abc + 2$ . * $f(u) = 3u + 2$ . (The instructor notes that "math humor is hard to get," referencing the spelling of $f(u)$ ).

General Rule: The domain consists of all suitable replacements for the input ( $x$ ) that do not cause the function to "go haywire."
Polynomials: For any polynomial where you add, subtract, multiply, and raise to powers (e.g., $f(x) = 3x + 2$ ), the domain is always all real numbers: $(-\infty, \infty)$ .
The Three "Problem Children" of Domains: 1. Rational Functions (Fractions): The denominator cannot be zero ( $0$ ). * Example: $f(x) = \frac{x+7}{2x-4}$ . Factor the denominator: $2(x - 2)$ . The value $x = 2$ makes the denominator zero and must be excluded. * Domain: $x \neq 2$ or $(-\infty, 2) \cup (2, \infty)$ . 2. Square Root Functions (Even Roots): You cannot have a negative number under the radical (radicand). * Example: $h(x) = \sqrt{x-3}$ . The radicand must be greater than or equal to zero ( $x - 3 \geq 0$ ). * Smallest value: $x = 3$ . * Domain: $[3, \infty)$ . 3. Logarithmic Functions: These also have restricted domains (to be discussed later).

Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph more than once. * A circle fails this test because a vertical line can intersect it at two points (one top, one bottom).
Determining Domain and Range from a Graph: * Domain: Look at the horizontal extent of the graph from left to right. * Range: Look at the vertical extent of the graph from bottom to top.
Graph Examples: * Case A (Endpoints): A graph starting at $x = -3$ and ending at $x = 5$ with a bottom at $y = -4$ and a top at $y = 6$ has a Domain of $[-3, 5]$ and a Range of $[-4, 6]$ . * Case B (Arrows): A graph with arrows pointing left and right that bottoms out at $y = 2$ has a Domain of $(-\infty, \infty)$ and a Range of $[2, \infty)$ .

Standard Form: $f(x) = mx + b$ . * $m$ is the Slope (change in $y$ over change in $x$ , or rise over run). * $b$ is the value used for the y-intercept (the point is explicitly $(0, b)$ ).
Graphing Example: $y = \frac{2}{3}x - 2$ : 1. Identify the y-intercept: $(0, -2)$ . 2. Identify the slope: $\frac{2}{3}$ (Up $2$ units, Right $3$ units). 3. Plot the anchor point $(0, -2)$ , then count up $2$ and over $3$ to find the next point.
Point-Slope Formula: Used when you know one point $(x_1, y_1)$ and the slope $m$ . * $y - y_1 = m(x - x_1)$ . * Example: Point $(3, -4)$ , Slope $m = 3$ . * $y - (-4) = 3(x - 3)$ . * $y + 4 = 3x - 9 \rightarrow y = 3x - 13$ .
Slope Formula for Two Points: * $m = \frac{y_2 - y_1}{x_2 - x_1}$ . * Example: Points $(3, 2)$ and $(7, 10)$ . * $m = \frac{10 - 2}{7 - 3} = \frac{8}{4} = 2$ .

Standard Form: $f(x) = ax^2 + bx + c$ ( $a \neq 0$ ).
Shape: The graph is a parabola. If $a > 0$ , it opens up. If $a < 0$ , it opens down.
The Vertex: The most important point on the parabola. * X-coordinate: $x = \frac{-b}{2a}$ . * Y-coordinate: Substitute the x-coordinate back into the function: $f(\frac{-b}{2a})$ .
Example Calculation ( $f(x) = x^2 - 8x + 12$ ): * Vertex x: $\frac{-(-8)}{2(1)} = 4$ . * Vertex y: $(4)^2 - 8(4) + 12 = 16 - 32 + 12 = -4$ . Vertex is $(4, -4)$ . * Y-intercept: Set $x = 0 \rightarrow y = 12$ . * X-intercepts: Set $y = 0 \rightarrow x^2 - 8x + 12 = 0$ . Factor into $(x - 6)(x - 2) = 0$ . X-intercepts are $(2, 0)$ and $(6, 0)$ .

Fundamental Purpose: Calculus is a tool for analyzing nonlinear functions, as linear functions are already fully understood through algebra.
The Three Main Problems in Calculus: 1. Limits: Analyzing what happens when you get "really, really close" to a particular number without necessarily reaching it. 2. The Tangent Problem (Derivatives): Finding the "slope of a curve." Since curves change direction, this refers to the slope of a line tangent to the curve at a specific point. 3. The Area Problem (Integrals): Finding the area underneath a curve. While areas for rectangles or trapezoids (linear boundaries) are easy, curves require calculus to measure the space between the function and the axis.

Student Question on Infinity: "When you have the either negative infinity or positive infinity, is it always a parenthesis around it?" * Response: Yes. You cannot include infinity ( $\infty$ ) because it goes on forever and ever; it is not a finite number you can reach.
Logistics: The instructor noted technical issues with MyMathLab due to class iteration changes. Students are encouraged to register between now and Wednesday.
Safety Joke: The instructor jokingly warned students that he would "hurl this marker as a projectile" if they incorrectly separated terms under a square root.