Lesson 2 — Understanding Functions

Overview

A function is one of the most fundamental ideas in mathematics — it’s the heartbeat of algebra, calculus, and beyond.
In everyday language, a function is simply a rule that links every input to exactly one output.

You’ve seen relationships before: a person and their fingerprint, a car and its VIN number, a student and their ID.
Each of these represents a function-like pairing: one input → one unique output.

This lesson explores how to describe, graph, and analyze these mathematical relationships through domain, range, notation, and behavior.

1. What is a Function? — The Rule of Uniqueness

A function is a special kind of relation between two sets:
each input value (from the domain) is mapped to exactly one output value (in the range).

Formal Definition

A relation ( f ) is a function if for every input ( x ), there exists only one output ( f(x) ).

[
x \mapsto f(x)
]

For example:
[
f(x) = x^2
]
means “take any ( x ), square it, and that’s your output.”

Function vs. Non-Function

Relation Is it a Function? Why ( f(x) = x^2 ) ✅ Yes Each ( x ) gives one ( y ) ( y^2 = x ) ❌ No Each ( x ) gives two possible ( y ) values (+√x and −√x)

Visual Test: The Vertical Line Test

If any vertical line crosses the graph more than once, it fails — it’s not a function.

2. Domain and Range — The Territory of a Function

Every function operates within a domain (its input set) and produces results in a range (its output set).

Domain (Input Set):

All the values of ( x ) that keep the function defined — no division by zero, no negative under even roots, etc.

Examples:

  • ( f(x) = \frac{1}{x-2} ) → domain excludes ( x = 2 )

  • ( g(x) = \sqrt{x+5} ) → domain is ( x \ge -5 )

Range (Output Set):

All possible ( y )-values the function produces.

Examples:

  • ( f(x) = x^2 ) → range ( y \ge 0 )

  • ( g(x) = \sin(x) ) → range ( -1 \le y \le 1 )

Notation

We can express this relationship as:
[
f: \text{Domain} \rightarrow \text{Range}
]

Example:
[
f: \mathbb{R} \rightarrow \mathbb{R}_{\ge 0}, \ f(x) = x^2
]
means ( f ) takes real numbers and outputs nonnegative reals.

3. Function Notation and Evaluation

Function notation compactly expresses the rule.
If ( f(x) = 2x + 3 ), then ( f(4) ) means “plug in 4 for x.”

[
f(4) = 2(4) + 3 = 11
]

Substitution Examples

  1. ( f(x) = x^2 - 5x + 6 )
    Find ( f(3) ):
    [
    f(3) = 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0
    ]

  2. If ( f(x) = 2x + 1 ) and ( g(x) = x^2 ),
    then ( g(f(x)) = (2x + 1)^2 = 4x^2 + 4x + 1 ).

This composition—feeding one function into another—creates the backbone for transformations and calculus operations later on.

4. Increasing and Decreasing Behavior

A function’s behavior describes how its outputs change as inputs increase.

Definitions:

  • Increasing: As ( x ) increases, ( f(x) ) increases.
    If ( x_1 < x_2 ), then ( f(x_1) < f(x_2) ).

  • Decreasing: As ( x ) increases, ( f(x) ) decreases.
    If ( x_1 < x_2 ), then ( f(x_1) > f(x_2) ).

  • Constant: ( f(x) ) stays the same no matter ( x ).

Example:

[
f(x) = x^2
]

  • Decreasing on ( (-\infty, 0] )

  • Increasing on ( [0, \infty) )

You can determine intervals of increase/decrease visually (by looking at the slope) or analytically (using derivatives later in Calculus).

Graphical Intuition

Imagine walking along the curve from left to right:

  • Uphill = increasing

  • Downhill = decreasing

  • Flat = constant

5. Even and Odd Functions — The Symmetry of the Graph

Functions have personalities — some are symmetric, some aren’t.

Even Functions

A function is even if it mirrors perfectly across the y-axis.

[
f(-x) = f(x)
]

Examples:

  • ( f(x) = x^2 )

  • ( f(x) = \cos(x) )

Graphical Symmetry:
Reflect the graph across the y-axis — it stays the same.

Odd Functions

A function is odd if rotating the graph 180° about the origin doesn’t change it.

[
f(-x) = -f(x)
]

Examples:

  • ( f(x) = x^3 )

  • ( f(x) = \sin(x) )

Graphical Symmetry:
Symmetric about the origin.

Neither Even nor Odd

Most functions are neither.
Example: ( f(x) = x^2 + x )

[
f(-x) = (-x)^2 + (-x) = x^2 - x \neq f(x) \text{ and } \neq -f(x)
]

So, it’s neither.

6. Types of Symmetry — Seeing Patterns

Symmetry tells us what makes a graph elegant or predictable.

Type Algebraic Test Graphical Effect Even ( f(-x) = f(x) ) Reflects across y-axis Odd ( f(-x) = -f(x) ) Rotational symmetry about origin Neither Neither condition met No symmetry

Real Examples:

  • The absolute value function ( f(x) = |x| ) is even — mirror symmetry.

  • The cubic function ( f(x) = x^3 ) is odd — rotational symmetry.

  • The exponential function ( f(x) = e^x ) is neither — only approaches 0 for negative ( x ).

7. Function Representation — The Four Faces

A function can appear in multiple forms, each telling a different story.

Representation Description Verbal A sentence: “Take a number, multiply by 2, then add 3.” Algebraic ( f(x) = 2x + 3 ) Numerical Table of values: pairs of ( x ) and ( f(x) ) Graphical Plot showing how ( f(x) ) changes visually

Understanding all four forms helps you transition smoothly between symbolic and geometric thinking — a core skill in Pre-Calculus.

8. Real-World Connection

Functions describe patterns and predict outcomes everywhere:

Field Example Function Physics ( h(t) = -16t^2 + v_0t + h_0 ) — projectile height Finance ( A(t) = P(1 + r)^t ) — compound interest Biology ( N(t) = N_0 e^{kt} ) — population growth Computer Science Hash functions mapping input → output

Every formula that models a process or behavior is a function. Understanding domain, range, and symmetry lets you interpret those models.

9. Common Function Families

A brief look ahead at the kinds of functions you’ll master:

Function Equation Key Feature Linear ( f(x) = mx + b ) Constant slope Quadratic ( f(x) = ax^2 + bx + c ) Parabola shape Cubic ( f(x) = ax^3 + bx^2 + cx + d ) S-shaped Absolute Value ( f(x) = x Exponential ( f(x) = a^x ) Rapid growth/decay Logarithmic ( f(x) = \log_a x ) Slow growth Rational ( f(x) = \frac{p(x)}{q(x)} ) Vertical/horizontal asymptotes Trig ( f(x) = \sin(x), \cos(x) ) Periodic behavior

10. Summary Table

Concept Definition Example Function One input → one output ( f(x) = 3x - 2 ) Domain All valid inputs For ( f(x)=1/x ), domain ≠ 0 Range All outputs produced For ( f(x)=x^2 ), range ≥ 0 Increasing ( f(x_1) < f(x_2) ) as ( x_1 < x_2 ) ( f(x) = e^x ) Decreasing ( f(x_1) > f(x_2) ) as ( x_1 < x_2 ) ( f(x) = -x ) Even ( f(-x) = f(x) ) ( f(x)=x^2 ) Odd ( f(-x) = -f(x) ) ( f(x)=x^3 )

Concept Reflection

Functions are not just rules—they’re relationships that connect the world’s quantities. Whether tracking a rocket’s flight or predicting market trends, functions describe change, growth, and balance. The more fluently you can manipulate domains, interpret graphs, and recognize symmetry, the more mathematics becomes a visual, living system instead of memorized equations.