CAT 2025 Lesson : Functions - Concepts & Cheatsheet

1) A function is a relationship, where every input has exactly one output. Functions exist only for One-to-One and Many-to-One relationships.

2.1) Domain is the set of all possible inputs or $x$-values.
2.2) Co-domain is the set of possible outputs or values of $f(x)$ that is defined by us.
2.3) Range is the set of outputs that are actually generated for the given inputs or domain. These are the actual values that $f(x)$ can possible take.

3.1) Onto Functions: Every element in the co-domain is related to at least one value in the domain. In this case, the co-domain is the range.
3.2) Into Functions: There is at least one element in the co-domain that is not related to a value in the domain. In this case, while the range is a subset of the co-domain, it has fewer values than the co-domain.

4) Function of 2 Sets: A function defined from Set A to a Set B is one wherein every element in A is related to exactly one element in B.

5.1) Even Function is one where $f(x) = f(-x)$.
5.2) Odd Function is one where $f(x) = -f(x)$.

6.1) Test for Function: Any vertical line intersects the graph of the expression at at most 1 point.
6.2) Number of inputs: Number of intersection points of a horizontal line with the expression.

7.1) Greatest integer function of x, also denoted as $[x]$, returns the greatest integer less than or equal to $x$.
7.2) Least integer function of $x$, returns the least integer greater than or equal to $x$.

8) To find the inverse of a function, please adhere to the following
(a) Replace $f(x)$ with $y$.
(b) Express $x$ in terms of $y$.
(c) In thee equation, replace $x$ with $f^{-1}(x)$ and $y$ with $x$.