CAT 2025 Lesson : Functions - Box & Composite Functions

6.Special Types of Functions

6.1 Greatest/Smallest Integer Functions

These are also called step functions as they look like steps when graphically plotted.

Greatest integer function of $x$, also denoted as $\bm{[x]}$, returns the greatest integer less than or equal to $x$. For instance, $[1] \space = \space 1 \space , \space [1.2] \space = 1 \space , \space [1.99] \space = \space 1, [2] \space = \space 2$ This is also called a floor function.
Least integer function of $x$, returns the least integer greater than or equal to $x$. For instance, LIF($1$) $= \space 1 \space , \space$ LIF($1.2$)$\space =\space 2 \space , \space$ LIF($1.99$) $\space = \space 2$, LIF($2$) $\space = \space 2$ This is also called a ceiling function.

6.2 Nested & Composite Functions

In nested functions, we will given a function $f(x)$ and asked to find $f(f(x))$. This is a function within a function.

In composite functions, we will be given $2$ or more functions, say $f(x)$ and $g(x)$. We will then be asked to find $f(g(x))$, etc.