Precalculus Fundamentals - Functions 📐

Introduction

• Course Purpose: Build a foundational understanding for calculus.
• Focus: Functions, to prepare for more advanced math concepts.
• Objective: Understand what a function is, how to evaluate them, and find their domain and range.

Key Concepts of Functions

Definition of a Function

• Basic Definition: A function maps one input to one output.
• Inputs and Outputs: Generally, inputs are denoted as X and outputs as Y or f(x).
• Independent Variable (X): The value you plug into the function, which is your choice.
• Dependent Variable (Y or f(X)): The value you get out from the function, dependent on the input.

Function Characteristics

• One-to-One vs. Basic Function: Basic functions map each input to one output. One-to-one functions also ensure each output is unique.
• Graphing: Graphing non-function relationships can be complex as an input may give multiple outputs.

Notation

• Y and f(X): Both denote the output value of a function. Different letters (like G or H) can be used to name functions for clarity.

Practical Example: Job Hours and Pay

• Analogy: Number of hours worked (X) determines pay (Y). This simple relation shows how an input (hours) maps to one output (pay).
• Non-Function Case: If identical inputs result in different outputs (e.g., same hours, different pay), it's not a function.

Domain and Range

• Domain: The set of all possible input values (X-values) for the function that result in real outputs.
• Range: The set of all possible output values (Y-values) the function can produce.

Identifying Functions with Examples

Simple Sequences

• Look at unique inputs (X-values). If each input has a unique output, it's a function.

Algebraic Representation

• Solved for Y: If an equation is solved for Y, it's easier to identify function properties.
• Potential Issues: Notation like ± (plus/minus) can indicate non-functions as they map one input to multiple outputs.

Examples

1. Simple Sequence Example

   • Inputs: -2, -1, 0, 1
   • Outputs: 16, 4, 3, 4
   • Conclusion: Function (each input has one output).

2. Non-Function Sequence Example

   • Inputs: -2, 3, 5, -2
   • Outputs: 5, 7, 9, 6
   • Conclusion: Non-Function (same input, different outputs).

Solving for Y in Algebraic Functions

Process

1. Isolate Y: Simplify equations so Y is alone on one side.
2. Identify Notation Issues: Look out for ± signs that indicate non-function characteristics.
3. Factor: For equations with Y on multiple terms, factorize to solve for Y.

Practical Example

1. Basic Linear Function

   • Equation: -7x + 5 = y
   • Conclusion: Each input of X results in one output for Y.

2. Quadratic Form

   • Equation: y = x^2
   • Conclusion: Function as each input maps to one unique output.

3. Square Root Example

   • Equation: ±√(3 - x)
   • Conclusion: Non-Function due to ± indicating non-unique outputs.

4. Solving Complex Functions

   • Example: 3y^2 = 2x^2 + 1
     • Steps: Isolate Y, take square roots, recognize ± makes it a non-function.

Conclusion

• Understanding Functions: Essential for graphing and preparing for advanced math concepts.
• Key Points: Each input must map to one unique output to qualify as a function.
• Next Steps: Practice identifying functions from algebraic expressions and real-life scenarios. Explore domain and range further in coming sessions.

---

Study Tip: Review the examples frequently and practice solving for Y to identify functions effectively.