1.1 Definition and Concepts of Functions

1. Introduction

Functions are fundamental concepts in mathematics and science, serving as the building blocks for modeling relationships between variables. They play a crucial role in various fields, from physics and engineering to economics and data science [Stewart, 2015].

2. Definition of a Function

A function $f$ from a set $A$ to a set $B$ is a rule that assigns to each element $x$ in $A$ exactly one element $y$ in $B$. We write this as:

$f: A \to B$

Here, $A$ is called the domain of the function, and $B$ is called the codomain. The set of all outputs $y$ is called the range of the function, which is a subset of the codomain [Spivak, 1994].

3. Key Concepts

• Input and Output: The input of a function is often denoted as $x$, while the output is denoted as $y$ or $f(x)$.
• Notation: Functions are often written in the form $y = f(x)$, where $f$ is the name of the function.
• Graph: The graph of a function is the set of all points $(x, y)$ in the Cartesian plane such that $y = f(x)$.
• One-to-One (Injective): A function is one-to-one if each element in the codomain is paired with at most one element in the domain.
• Onto (Surjective): A function is onto if every element in the codomain is paired with at least one element in the domain.
• Bijective: A function that is both one-to-one and onto.

4. Types of Functions

• Linear Functions: Of the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Quadratic Functions: Of the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.
• Exponential Functions: Of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.
• Logarithmic Functions: The inverse of exponential functions, of the form $f(x) = \log_a(x)$.
• Trigonometric Functions: Such as sine, cosine, and tangent.

5. Interactive Example: Linear Function

Let's explore a linear function of the form $f(x) = ax + b$. Use the sliders to adjust the values of $a$ (slope) and $b$ (y-intercept) and observe how the graph changes.

6. Real-World Application: Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering that describes the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit. It can be expressed as a linear function:

$V = IR$

Where:

• $V$ is the voltage across the conductor (measured in volts, V)
• $I$ is the current through the conductor (measured in amperes, A)
• $R$ is the resistance of the conductor (measured in ohms, Ω)

This linear function demonstrates how the concepts we've discussed apply in a practical engineering context:

• Domain: The current $I$ (typically non-negative real numbers)
• Codomain: The voltage $V$ (all real numbers)
• Slope: The resistance $R$ (determines how quickly voltage increases with current)
• Y-intercept: In this case, it's 0, as no current means no voltage drop

7. Function Composition

Function composition is the process of combining two or more functions to create a new function. If $f$ and $g$ are two functions, their composition is denoted as $(f \circ g)(x)$ or $f(g(x))$, which means "apply g first, then apply f to the result" [Stewart, 2015].

Mathematically, this is expressed as:

$(f \circ g)(x) = f(g(x))$

For example, if $f(x) = x^2$ and $g(x) = x + 1$, then:

$(f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2$

8. Conclusion

Functions are powerful mathematical tools that allow us to model relationships between variables, make predictions, and solve complex problems across various scientific and engineering disciplines. From the simple linear functions we explored interactively to more complex compositions, functions provide a language for describing and analyzing the world around us [Stewart, 2015; Boas, 2006; Spivak, 1994].

As you continue your studies, you'll encounter more advanced function concepts, such as limits, derivatives, and integrals, which form the foundation of calculus and higher mathematics. These tools will enable you to tackle even more complex problems in science, engineering, and beyond.