Piecewise Functions examples aid students to understand the steps involved in solving Piecewise functions.
A function can be termed as a mechanism where a number is taken from you and giving or returning a number back to you. This normally is an arithmetic operations sequence in a math class. Ignorantly that's how the function behaves normally, doesn't matter what number is given to it applying the same process. Piecewise function is more realistic in behavior. It does look at the number before deciding what it has to do based on the number. The absolute value function is the first example most people see. The function has to determine before deciding what to do with it whether the number it is assigned is positive or negative. The absolute value leaves it alone if the number is already positive. The absolute value will have to change its sign if the number is negative.
In Pre calculus,a function can be represented by a single equation. But in the real life, we use more than one equation, each for a particular set of domain, to represent a particular situation. Such functions are called “Piecewise Functions”.
This function says f(x) = x – 0.5, when x is less than 1, and f(x) = x + 1, when x is greater than 1.
Piecewise Functions Examples
The following is an Piecewise Functions examples and this pre calculus problem explains how to solve this kind of functions:
Consider the Piecewise Functions examples:
Evaluate f(x) when (a) x = 1 (b) x = 0 and (c) x = -1
Solution for the given Piecewise Functions examples: (a) x = 1,
f(x) = x + 2
f(1) = 1 + 2 = 3
(b) x = 0,
f(x) = x² - 1
f(0) = 0² - 1 = -1
(c ) x = -1
f(x) = x² - 1
f(-1) = (-1)² - 1= 0
Hence the Piecewise Functions examples are solved.
Graphing a Piecewise Function
When graphing Piecewise functions, we have to consider each piece and treat it as a separate function.
Break Point: The place on the graph, where the function changes from one piece to another piece.
While graphing, we have to graph the left portion of the graph upto the break point and end it as a solid dot or an open dot depending upon the condition given. We have to graph the right side of the graph with a solid or open dot, again depending on the given condition. Here are Piecewise Functions examples including graphing:
Graph the following Piecewise Functions.
Break Point : Observations
Break Point Observations:
- Break point is at x = 0.
- There is an open dot at (0, 2). When we substitute x = 0, in f(x) = x + 2, we get f(x) = 2 and hence an open dot is placed at (0, 2) which is the point of culmination of the left portion of the graph.
- There is a solid dot at (0, 0). When we substitute x = 0 in f(x) = x, we get 0, and hence there is a solid dot placed at (0, 0) as the relationship for the right portion of the graph is ?.
The same procedure can be used even if the Piecewise Functions examples given have three pieces or more. If the piecewise function given has 3 pieces, then there will be two break points. We must take utmost care to place the solid and open dots appropriately.
A step function is a function, whose graph is a set of lines which appear like steps. The Greatest Integer and Smallest Integer Functions are good examples of Step Functions.
Domain of Step Piecewise Function is divided into a number of equal intervals and in each interval, the Range is a constant. This means that within an interval, the value of the Step Function does not change.
Greatest Integer Function
This pre calculus homework help problem on piecewise functions
The intervals of greatest integer function are [k,k+1) and f(x) = k when k ? x < (k+1) where k is an integer.i.e.
f(x) = 0 for 0?x<1
f(x) = 1 for 1?x<2
f(x) = 2 for 2?x<3 and so on.
Now let us graph this function.
There are a number of real life situations which can be well represented by Step functions. The Greatest Integer Function is also called as the Floor Function. For every real value of x, the greatest integer is less than or equal to x.
Similarly another example of Step function is the Smallest Integer function, which is also called as the Ceiling function.