A function can be made with many pieces. For example, after some certain values, the function can change its shape and once it changes its shape that means a change in the equation. The pieces have no specific limit, which means that a function can have many pieces. Since the function has pieces that is why we call it a piecewise function.  The piecewise functions are functions defined by different criteria according to the intervals being considered. For example:

y = \left\{\begin{matrix} {x}^{2} \quad if \quad x<2\\4 \quad if \quad x>2 \end{matrix}\right

Looking at the above graph, we can conclude a few things. Firstly, we observed that there are two different shapes, which means that we are dealing with two different pieces but within the same domain. Secondly, the breakpoint is 2 which means that there is a shift in the equation at point x = 2. When x < 2, the graph was following the equation y = { x }^{ 2 } hence you will see the graph of equation y = { x }^{ 2 } from infinity to x = 2. However, at x = 2, things are about to change, the graph is following the equation y = 4, and since y = 4 has no limit, it will continue till infinity.

Domain of y: (- \infty, +\infty)

Floor Function

The floor function or greatest integer function gives the largest integer less than or equal to x. For example, you are asked to figure out the floor function of 3.67, now you need to find the greatest integer that is less than 3.67. Some of you might be thinking the largest integer less than 3.67 will be 3.66 and that is where you are wrong, we are talking about integers here. Hence, the largest integer which is less than 3.67 will be 3.

y = \left \lfloor x \right \rfloor

x 0 0.5 0.9 1 1.5 1.9 2
y = \left \lfloor x \right \rfloor 0 0 0 1 1 1 2

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Ceiling Function

The ceiling function or least integer function gives the smallest integer greater than or equal to x. For example, we will take the above example again but with a small change, this time you were asked to figure out the ceiling function of 3.67. The answer is 4 because 4 is the least integer which is greater than 3.67.

y = \left \lceil x \right \rceil

x 0 0.5 0.9 1 1.5 1.9 2
y = \left \lceil x \right \rceil 1 1 1 2 2 2 3

y = \left \{ x \right \}

\left \{ x \right \} = x - \left \lfloor x \right \rfloor

x 0 0.5 0.9 1 1.5 1.9 2
\left \{ x \right \} 0 0.5 0.9 0 0.5 0.9 0

y = sgn(x)

y = \left\{\begin{matrix} -1 \quad if \quad x < 0 \\ 0 \quad if \quad x = 0 \\ 1 \quad if \quad x > 0 \end{matrix}\right

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.