Functions, Limits and Continuity Overview

This topic covers four essential areas that build on each other. Functions and models form the foundation, followed by domain and range concepts, then limits using numerical and graphical methods, and finally algebraic limits and continuity.

Think of this as your roadmap to understanding how mathematical relationships work. Each section prepares you for more advanced calculus concepts you'll encounter later.

Functions and Models Basics

Ever wondered what makes something a function? It's simpler than you think - a function is just a special type of relation where each input has exactly one output. Unlike regular relations, no two ordered pairs can have the same first element.

Here's the key difference: if you can draw a vertical line through a graph and it touches the curve more than once, it's not a function. For example, y = x² is a function, but x = y² isn't because it fails the vertical line test.

Functions use special notation like f(x) = x², which is just another way of writing y = x². This notation makes it easier to work with and substitute values.

Quick Tip: Remember the vertical line test - it's your go-to method for identifying functions on graphs!

Function Evaluation Practice

Function evaluation is like following a recipe - you substitute the given value for x and calculate the result. With f(x) = 3x + 5, finding f(4) means replacing every x with 4: f(4) = 3(4) + 5 = 17.

The real power comes when you work with variables. For f(a), you get 3a + 5, and for f$a + h$, you get 3$a + h$ + 5 = 3a + 3h + 5.

This skill becomes crucial for more advanced topics like derivatives, so practice until substituting values feels automatic.

Pro Tip: Always work step-by-step and double-check your arithmetic - small errors here lead to big problems later!

Quadratic Function Evaluation

Quadratic functions like f(x) = 3x² + 2x - 7 require extra care because you're dealing with squared terms. When evaluating f(4), you calculate 3(4)² + 2(4) - 7 = 48 + 8 - 7 = 49.

For variables, f(a) = 3a² + 2a - 7, and f(5a) = 3(5a)² + 2(5a) - 7 = 75a² + 10a - 7. Notice how (5a)² becomes 25a², then multiplied by 3 gives 75a².

The key is handling the order of operations correctly - squares first, then multiplication, then addition and subtraction.

Watch Out: (5a)² = 25a², not 5a². This is one of the most common mistakes students make!

Difference Quotients

The difference quotient $f(x + h) - f(x)$/h might look intimidating, but it's just measuring how much a function changes. For f(x) = 2x - x², you first find f$x + h$ = 2$x + h$ - $x + h$².

Expanding gives f$x + h$ = 2x + 2h - x² - 2xh - h². Then subtract f(x) to get -2h - 2xh - h². Finally, divide by h to get -2 - 2x - h.

This concept becomes the foundation for derivatives in calculus - you're essentially finding the slope of a curve at any point.

Key Insight: The difference quotient measures the average rate of change, which leads directly to instantaneous rate of change in calculus!

Graphing Parabolas

Graphing quadratic functions like f(x) = 2 - x² starts with recognizing the basic shape. Since the coefficient of x² is negative, this parabola opens downward with vertex at (0, 2).

Create a table of values to plot points: when x = -2, y = 2 - 4 = -2; when x = 0, y = 2; when x = 2, y = -2. The symmetry around the vertex makes graphing easier.

The vertex form y - 2 = -$x - 0$² clearly shows the vertex at (0, 2) and confirms the downward opening.

Memory Aid: Negative coefficient of x² means sad parabola (opens down), positive means happy parabola (opens up)!

Piecewise Functions

Piecewise functions are like mathematical chameleons - they change their behavior based on the input value. For the given function, you have three different rules for three different intervals.

When x ≤ 0, f(x) = 4 (horizontal line). For 0 < x ≤ 2, f(x) = 4 - x² (downward parabola starting from (0,4)). When x > 2, f(x) = 2x - 6 (straight line with slope 2).

Pay attention to the inequality symbols - they tell you whether to include endpoints with solid or open dots.

Graphing Tip: Draw each piece separately, then check the endpoints to see if they connect or have breaks!

Special Piecewise Cases

Sometimes piecewise functions have unusual definitions, like a single point with one value and everywhere else with another. Here, f(-2) = 1 specifically, while f(x) = 2 - x for all other x values.

Graph y = 2 - x normally (a straight line), but place an open circle at (-2, 4) since that point isn't included. Then add a solid dot at (-2, 1) to show the special case.

This creates a function with a "hole" in the line and an isolated point above it.

Important: The solid dot shows where the function actually is, while the open circle shows where it would be on the main rule!

Domain and Range Introduction

Domain and range are about understanding what goes in and what comes out of a function. The domain consists of all possible input values $x-values$, while the range includes all possible output values $y-values$.

Interval notation provides a clean way to express these sets. You'll use brackets [ ] for included endpoints and parentheses ( ) for excluded endpoints.

This topic builds directly on your function knowledge and prepares you for more advanced concepts in calculus.

Think of it this way: Domain is like the ingredients you can use, range is like the dishes you can actually make!