Greatest Integer Function

The function, or rule which produces the "greatest integer less than or equal to the number" operated upon, is known as greatest integer function. It is denoted by symbol ⌊⌋ , i.e. ⌊x⌋≤x. It is also known as floor of x. Thus ⌊4.5778⌋=4 , ⌊0.75⌋=0 , ⌊-8.7275⌋=-9.

The Graph of the greatest integer function lies on or below the line y = x so it provides an integer floor for x.

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Piecewise Functions

A piecewise function is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain (a subdomain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial functions: a function that is a polynomial on each of its subdomains, but possibly a different one on each.

Note that, in this case, we have three different “formulas” for f(x): -x + 1, 2 and x²- 4. But remember, with a function, when you evaluate f(x), you should get ONE ANSWER—not three. Otherwise, you don’t have a function.

How do you know which answer you should have (i.e., which formula you should use)? The “formula” is chosen using the right-side of the piecewise-defined function (the “if …” part).

For example, suppose we wanted to find f( − 3). Since − 3 < − 1, we will use the first “formula”:

f( − 3) = - (-3) +1 = 4.

If we want to find f(5), since 5 > 3, we use the third “formula”:

f(5) = 5² - 4 = 21.

What about finding f(1)? Note that in the interval (-1,3) the value of function is 2.

The following applet shows the graph of piece wise function 

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Shifting Graphs

In this applet we will explore how to change an equation to shift its graph up or down or left or to the right or left. This can help us spotting familiar graphs in new positions. This will also help us graph unfamiliar equations more quickly.

To shift the graph of a function y=f(x) straight upwards, we add a constant to the right hand side e.g. by adding 2 to the right hand side of the formula y = x² we get y = x²+2 and the graph is shifted up by 2 units.

To shift the graph of equation y = f(x) straight down, we subtract we add a negative constant to the right hand side of the formula y = f(x)

To shift the graph of y = f(x) left or right , we add positive or negative constant to x respectively e.g. if by adding 2 to x in y = x² to get y = (x+2) ² , shifts the graph 2 units to the left.

In general we can say

•  y = f(x) + b , shifts the graph up by b units if b > 1 or shifts it down by |b| units if b <0 
•  y = f(x-a) , shifts the graph right a units if a > 0 or shifts it left |a| units if a<0. 

 Right Click any where on the graph and select ‘Trace On’ , then drag sliders ‘a’ or ‘b’.


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Radical Functions

A radical function is any function that contains a variable inside a root. This includes square roots, cubed roots, or any nth root , for example f(x) = √(x+5) , g(x) = ∜(x-2) etc.

Here we will discuss about functions involving independent variable inside a square root sign or square root functions. In general the form of a square root function is                 f(x)=a√(x-b)+c , where a , b and c are some real numbers.

Graph of such a function is shown in the applet below.

To find domain of a square root function , the term inside the radical must be equal to or greater than zero, otherwise it is undefined. This means that only the x values that make the term inside the radical positive are defined and in the domain. 
For example , f(x) = √(x-4) + 3.

Since (x-4) is inside the radical, the domain lies on all the points where x makes (x-4) greater than or equal to zero. 
       x-4 ≥ 0
       x ≥ 4 

So the domain of the function is [4, ∞). 

The range of the function is then all the points of the y-axis that we get by putting x values of the domain. Let us start at the point x=4 and put it into the equation. f(x)=√(4-4)+3=3.

Now it can be easily concluded that by putting in any number greater than 4 for x we get f(x) larger than 3, so the smallest number in the range is 3. Thus we see that for any x value above 4 the function is defined. Therefore the range is [3,∞).


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Even - Odd Functions

Every function can be classified as an even function, an odd function, or neither. Even functions have the characteristic that f (x) = f (-x). They are symmetrical with respect to the y-axis. A line segment joining the points f (x) and f (-x) will be perfectly horizontal , shown by dotted blue line in the applet below.

Odd functions have the characteristic that f (x) = - f (-x). They are symmetrical with respect to the origin. A line segment joining the points f (x) and f (-x) always contains the origin , shown by dotted red line in the applet below. 

Some of the most common even functions are y = k , where k is a constant, y= x² , and y = cos(x) . Some of the most common odd functions are y = x³ and y = sin(x) . Some functions that are neither even nor odd include y = x - 4 , y = cos(x) + 1.

Formal tests for symmetry: 
1. y – axis : replace x with –x , produces an equivalent equation 
2. x - axis : replace y with – y , produces an equivalent equation 
3. origin : replace x with –x and y with –y , produces equivalent equation. 

In the following applet use check box to select between odd or even functions. Also you can enter any function in the input box to see whether it is odd or even by checking the symmetry about y-axis or origin.

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