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)
There is also this idea of Relational Composition
We say that a function is an inverse of a function if for any in the domain of .
inverses are unique.
Note that the inverse of f is NOT the same thing as the reciprocal of f.
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. -> you want to write in terms of You know that , so just swap the variables, you have , so you get
In general, you let , so if (this is invertible), you have (so simply swap the variables from the original function), and solve for , you get . The final answer is
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
- 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.