📘 Functions of One Variable

A function is, to its core, is a rule, a relationship, a change, or a connection. A map that describes a strong relationship between two concepts, where there is no ambiguity of how these two objects relate to each other. Each input is related exactly to one output

Functions are the most useful tool to model Real-World phenomena. We can model it as a machine: you put something in, and you get something out.

• Height as related to age
• Cost of transport, related to weight, in the post office.
• Temperature as measured over time.
• Area of a circle as dependent on the Radius.

🔹 What Is a Function?

A function expresses a rule or law of correspondence between two quantities. Each value of the independent variable corresponds to exactly one value of the dependent variable.

In symbolic form:

$$y = f(x)$$

We read this as “y is a function of x” or “f of x.” Here, f names the rule that links input x to output y.

function(x) = y

𝕏 \xrightarrow{𝑓} 𝕐

𝑓:⠀𝕏 ⟶ 𝕐

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🔹 Independent and Dependent Variables

• $x$ is the independent variable:

  • The input, free within some domain.
  • 𝕏: Input Set, Domain.

• $y$ is the dependent variable:

  • The output, determined by applying f to x.
  • 𝕐: Output Set, Range, Codomain.

This distinction reflects a unidirectional dependency: $x \rightarrow f(x)$

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🔹 Ways to Represent a Function

A function may be expressed in several forms:

• 📊 Table of values — discretely listing input-output pairs
• 📈 Graph — plotting all $(x, f(x))$ on the Cartesian plane
• 🧮 Equation — an algebraic formula such as $f(x) = x^2 - 10$

Each representation encodes the same functional behavior through different mediums.

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🔹 Example

Given:

$x^2 - y = 10 \Rightarrow y = x^2 - 10$

This defines $y$ as a function of $x$. Every input $x$ yields a unique $y$. However, reversing roles:

$x^2 = y + 10 \Rightarrow x = \pm\sqrt{y + 10}$

Now $x$ is not a function of $y$: one input gives two outputs. This is a multiform relation.

In analysis, we often resolve such ambiguity by treating each branch (e.g., $+\sqrt{y+10}$, $-\sqrt{y+10}$) as a separate function.

🔹 Function Notation

The notation $f(x)$ denotes the value of function $f$ at point $x$. Crucially, this is not multiplication — $f$ is a rule, not a coefficient.

Multiple functions may be denoted by: $f(x), g(x), h(x), F(x), \varphi(x)$, etc.

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🔹 Domain and Range

• The domain of $f$ is the set of all permissible values for $x$.
• The range (or image) is the set of values $f(x)$ can attain.
• The codomain is the space into which $f$ maps, often $\mathbb{R}$.

Example:

• $f(x) = \sqrt{x} \Rightarrow \text{domain: } [0, \infty)$
• $g(x) = \frac{1}{x-3} \Rightarrow \text{domain: } \mathbb{R} \setminus {3}$

Formally:

$\text{Range}(f) = {f(x) \mid x \in \text{Domain}(f)}$

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🔹 Function as Machine

A function is like a machine:

flowchart LR
  X{Input x} --> F{{"Function f"}} --> Y>Output y]

Each input passes through the rule and emerges transformed.

• In a Mill-Function we'd get:

flowchart LR
  X{Wheat} -->F{{Mill}} --> Y>Flour]

Each input passes through the rule and emerges transformed.

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🔹 Key Properties

• Injective (One-to-one): different inputs produce different outputs
• Surjective (Onto): the range equals the codomain
• Bijective: both injective and surjective (invertible)

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🔹 Continuity

A function is continuous if you can zoom in on any point and find neighboring points without jumps. Formally:

$\forall \varepsilon > 0, \exists \delta > 0: |x - a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon$

Continuity is a local property with profound global consequences (e.g., Intermediate Value Theorem).

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🔹 Inverse Functions

A function $f$ has an inverse $g$ if:

$g(f(x)) = x \quad \text{and} \quad f(g(y)) = y$

This requires $f$ to be bijective. Graphically, $f$ and $f^{-1}$ are symmetric about the line $y = x$.

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🔹 Summary Table

Concept	Description
Function	Rule assigning one output to each input
Notation	$f(x)$: value of $f$ at input $x$
Domain	Permissible values of $x$
Range	Values actually taken by $f(x)$
Injective	One-to-one mapping
Surjective	Full coverage of codomain
Inverse	Exists if function is bijective

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