Functions

1. Definition / Introduction

A function (or mapping) $f$ 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 $f:A\to B$ and $y=f(x)$.

Formal: $f$ is a function $f:A\to B$ if $\forall x\in A\, \exists! \,y\in B$ such that $y=f(x)$.

Example: $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^2$ assigns to each real number its square.

2. Domain, Co-domain and Range

Domain of $f:A\to B$ is the set $A$ (all allowable inputs).
Co-domain is the set $B$ (the target set where outputs lie).
Range (image) of $f$ is $f(A)=\{f(x):x\in A\}\subseteq B$ (actual outputs).

Note: Range $\subseteq$ Co-domain; they may be equal or not.

3. Types / Classification of Functions

Below each type: brief definition, formula and a small plotted example (canvas).

3.1 Polynomial Function

Definition: $f(x)=a_nx^n+\dots+a_1x+a_0$, where $n$ is a non-negative integer.

Example: $f(x)=x^3-2x$

3.2 Algebraic Function

Definition: Built using algebraic operations (polynomials, roots). Example: $f(x)=\sqrt{x+1}$ (domain $x\ge-1$).

Example: $f(x)=\sqrt{x+1}$

3.3 Rational Function

Definition: Ratio of two polynomials $f(x)=\dfrac{p(x)}{q(x)}$, where $q(x)\neq0$.

Example: $f(x)=\dfrac{1}{x-1}$ (vertical asymptote at $x=1$).

3.4 Constant Function

Definition: $f(x)=c$ for all $x$ in domain.

Example: $f(x)=3$

3.5 Identity Function

Definition: $f(x)=x$.

Example: $f(x)=x$

3.6 Absolute Value Function

Definition: $f(x)=|x|$.

Example: $f(x)=|x|$

3.7 Signum Function

Definition: $\mathrm{sgn}\left({x}\right)=\begin{cases}{−\mathrm{1}}&{{x}<\mathrm{0}}\\{\mathrm{0}}&{{x}=\mathrm{0}}\\{\mathrm{1}}&{{x}>\mathrm{0}}\end{cases} $

Example: signum plot

3.8 Greatest Integer Function (GIF)

Definition: $\lfloor{x}\rfloor=\mathrm{greatest}\:\mathrm{integer}\:\leqslant\:{x}$. (Step function)

Example: $\lfloor{x}\rfloor$

3.9 Fractional Part Function

Definition: $\{x\}=x-\lfloor x\rfloor$, value in $[0,1)$.

Example: $\{x\}$

4. Algebraic Operations on Functions

If $f$ and $g$ are functions with domains where the operations make sense, we define:

Sum: $(f+g)(x)=f(x)+g(x)$
Difference: $(f-g)(x)=f(x)-g(x)$
Product: $(fg)(x)=f(x)\cdot g(x)$
Quotient: $\left(\dfrac{f}{g}\right)(x)=\dfrac{f(x)}{g(x)}$, provided $g(x)\neq0$
Composition: $(f\circ g)(x)=f(g(x))$ (domain where $g(x)$ lies in domain of $f$)

Example: Let $f(x)=x^2$, $g(x)=x+1$. Then

$(f+g)(x)=x^2+x+1$
$(fg)(x)=x^2(x+1)=x^3+x^2$
$(f\circ g)(x)=f(g(x))=(x+1)^2=x^2+2x+1$

5. Important Properties & Types (brief)

Injective (one-to-one): $f(x_1)=f(x_2)\Rightarrow x_1=x_2$.
Surjective (onto): Range $= $ Co-domain.
Bijective: Both injective and surjective → inverse function exists.
Even / Odd: $f(-x)=f(x)$ (even), $f(-x)=-f(x)$ (odd).

Example: $f(x)=x^3$ is odd and bijective (as $f:\mathbb{R}\to\mathbb{R}$). $f(x)=x^2$ is even and not injective on $\mathbb{R}$.

6. Rules

Domain restrictions: watch for denominators ($q(x)\ne0$), even roots ($x+1\ge0$), logs ($x>0$) etc.
To find range: algebraic manipulation, complete the square (for quadratics), or consider monotonicity.
Composition domain: $\operatorname{Dom}(f\circ g)=\{x\in\operatorname{Dom}(g):g(x)\in\operatorname{Dom}(f)\}$.

7. Examples

Find domain and range: $f(x)=\dfrac{1}{x-2}$. Domain: $x\ne2$. Range: $y\ne0$ (since $1/(x-2)$ attains every real except 0).
Composition: $f(x)=x^2,\ g(x)=x+1$. Then $(f\circ g)(x)=(x+1)^2$, domain all reals.
Inverse existence: $f(x)=x^3$ is bijective on $\mathbb{R}$; inverse $f^{-1}(x)=\sqrt[3]{x}$.

8. Quick reminders

Domain: inputs (left); Co-domain: potential outputs (right); Range: actual outputs (highlighted).
Composition needs the output of inner function to lie in domain of outer.
Operations require common domain (or restrict domain appropriately).