# Function

A function is something that maps an input (domain element) to a **unique** output (range element).

- If the output is not unique for a given input element, then we have a Relation

### Domain and Range

- The
**domain**of a function is the set of allowable values for the independent variable. - The
**range**is the set of possible values for the dependent variable.

### Composition of Functions (Function Composition)

$f∘g(x)=f(g(x))$ There is also this idea of Relational Composition

### Inverse Functions

We say that a function $f_{−1}$ is an *inverse* of a function $f$ if $f_{−1}(f(x))=x$ for any $x$ in the domain of $f$.

inverses are unique.

Note that the inverse of f is NOT the same thing as the reciprocal of f. $(f(x))_{−1}=f(x)1 =f_{−1}(x)$

**Invertibility**: Not every function possesses an inverse. For the inverse to be a true function, it must have a single output for every input. The requires that the original function is *one-to-one*, i.e. an Injective Function

in STAT206, to get an inverse, swap the two variables and solve. $F(x_{1})=u_{1}$ → you want to write in terms of $x_{1}$ $F_{−1}(u_{1})=x_{1}$ You know that $F(x)=1−e_{−x}$, so just swap the variables, you have $1−e_{−x_{1}}=u_{1}$, so you get $x_{1}=−ln(1−u_{1})$

In general, you let $x=f_{−1}(y)$, so if $f(x)=x =y$ (this is invertible), you have $y =x$ (so simply swap the variables from the original function), and solve for $y$, you get $y=x_{2}$. The final answer is $f_{−1}(x)=x_{2}$

### Classifications of Functions

- Even and Odd Function
- One-to-One and Onto Function
- Hyperbolic Function
- Piecewise Function
- Periodic Function
- Rational Function
- Trigonometric Function
- Exponential Function
- Logarithmic Function

Other:

**Algebraic function**: produced by taking sums, products, and quotients of roots of polynomials and rational functions- Any function that is not algebraic is called
**transcendental**. Exponential and trigonometric functions are examples.

### Related

### Categorization of Functions

From SE212, TODO add notes with Binary Relation