Piecewise functions Video transcript Select the piecewise function whose graph is shown below. Or I guess we should say to the right. I copied and pasted it so it's on the right now. So we have this piecewise continuous function.
Function examples A function is a mapping from a set of inputs the domain to a set of possible outputs the codomain. The definition of a function is based on a set of ordered pairs, where the first element in each pair is from the domain and the second is from the codomain.
Below, the domain is visualized as a set of spheres and the codomain as a set of cubes, so that the function machine transforms spheres into cubes. We often think of a function as taking a number as its input and producing another number as its output. But, we could make a function machine that operates on different types of objects, so a function is in no way limited to numbers.
To illustrate this fact, we start with examples that operate on objects other than numbers. Then, we turn to more traditional functions where the domain and codomain are sets of numbers.
For the purpose of making this example simple, we will assume all people have exactly one mother i. If, for example, we put Martin Luther King, Jr.
Number of children A function can output objects of a completely different type than the inputs, as suggested by the above picture where spheres enter the function machine and cubes come out.
Symbols The domain and codomain of a function could be sets of any type of objects. Such algebraic formulas are the way many people think of functions, though, as the above examples show, such a formula is not required.
To fully define a function, we need to specify the domain and range. If the domain and range are not specified, it is frequently safe to assume that the domain and range are the set of real numbers.
However, in most cases, we won't need to worry about such differences. For functions whose input is a number and output is a number, we can visualize the set of ordered pairs in terms of its graph. Any formula that unambiguously assigns an element in the codomain for each element in the domain will define a function.
We refer to such a formula as a piecewise formula, as it breaks the domain into pieces and uses a separate formula for each piece.
Other examples As suggested by the function machine metaphor, there's an endless variety to the types of functions you could define. You just need to come up with a collection of objects for the input, a collection of objects for the possible outputs, and decide what the function machine with spit out for each input object.
The input or output objects could even be sets containing many subparts.Step Functions and Piecewise Functions. In these lessons, we will learn how to graph and solve Step Functions and Piecewise Functions. Some functions are not easily written as a formula.
On a graph, a step function looks like a flight of stairs.
The graphs of step functions have lines with an open circle on one end and a closed circle on. Writing equations for piecewise functions: First take a given line and find the slope, then use the point slope formula to write an equation.
Then define the domain over which the function occurs. He determines which is the correct formula. Graphing nonlinear piecewise functions (Algebra 2 level) Graphs of nonlinear piecewise functions.
if I were to write it out, it looks like over this first interval.
So it looks like it's the square root of x plus 2 for negative 2 being less than x-- . Evaluating a piecewise function adds an extra step to the whole proceedings. We have to decide which piece of the function to plug-and-chug into. Since -3 is less than 2, we use the first function to evaluate x .
Interpolation by Splines KEY WORDS. interpolation, polynomial interpolation, spline. GOAL. Understand what splines are A simple piecewise polynomial fit is the continuous linear interpolating spline. For the data set x x 0 x 1 x n y f 0 f 1 f n where a= x 0 write S 1;n(x) = Xn i=0 f i.
This is definitely a start, however my main problem is creating a more general converter so that I don't have to write a special case for each possible piecewise expression.
– user Jun 27 '12 at