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)

$f\circ g(x)=f(g(x))$ 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. $(f(x){)}^{-1}=\frac{1}{f(x)}\ne f^{-1}(x)$

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 $f(x)=\sqrt{x}=y$ (this is invertible), you have $\sqrt{y}=x$ (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

• Curve Sketching
• Transfer Function

Categorization of Functions

From SE212, TODO add notes with Binary Relation