# Write a Piecewise Function for the Graph Below

Write a Piecewise Function for the Graph Below.

Contents

## Modeling a piecewise-defined part from its graph

## Building a piecewise office from its graph

In this lesson nosotrosâ€™ll look at piecewise-defined functions and how to write the equation of the definition of such a function given its graph.

A piecewise-divers part (also called a piecewise part) is a function thatâ€™s made upwards of different â€śpieces,â€ť each of which has its own â€śsub-functionâ€ť (its own algebraic expression) and its own â€śsub-domainâ€ť (its ain part of the domain of the entire piecewise part).

Weâ€™ll telephone call the â€śsub-functionâ€ť for each piece the

*function*

for that piece. A piecewise function is divers by giving the algebraic expression for the office for each piece and its domain. The domain of a piece of a piecewise office can be either an interval or simply a single bespeak.

The definition of a piecewise function is written in this class:

???f(x)=\begin{cases}\text{Function}_1 & \quad ten\text{-values}\\\text{Function}_2 & \quad x\text{-values} \\\text{Function}_3 & \quad 10\text{-values}\\\text{Function}_4 & \quad x\text{-values}\stop{cases}???

Of class a piecewise office doesnâ€™t need to have four pieces. Information technology can take anywhere from two pieces to an infinite number of pieces. Ordinarily, there are simply two or three.

Letâ€™s wait at an example of a definition of a piecewise function and how to graph the function.

???f(x)=\begin{cases}x^ii & \quad 10 <iii \\8 & \quad x = 3\\2x+four & \quad x > 3\end{cases}???

To graph a piecewise part, you graph each piece on its domain. Letâ€™s start by graphing the piece with function ???x^two??? and domain ???x<3???, which is (role of) a parabola that opens upwards and has its vertex at the origin.

The domain of the next piece is simply ???10=3???, and weâ€™re given that ???f(iii)=8???, so weâ€™ll plot the point ???(three,8)???. For our graph to be the graph of a function, we canâ€™t have 2 or more points with the same value of ???x???, and so weâ€™ll need to describe an open circle at the â€ścorrect ceaseâ€ť of the parabola, that is, at the signal with ???ten=three???, which has coordinates ???(3,3^two)???, or ???(3,9)???, to show that the betoken ???(3,9)??? isnâ€™t a point of the graph of this piecewise function.

Next we demand to graph the piece with office ???2x+4??? and domain ???x>3???, which is (part of) a line. When nosotros graph this piece of the function, weâ€™ll need to draw an open circumvolve at the bespeak of this line with ???x=3???, which has coordinates ???(3,2(3)+4)???, or ???(iii,10)???, because it isnâ€™t a indicate of the graph of this piecewise function.

You tin can as well write the equation of a piecewise function when youâ€™re given a graph.

## How to break apart the pieces of the piecewise role

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## Defining the piecewise function

**Case**

What is the definition of the piecewise function shown in the graph?

Weâ€™ll work on the graph from left to right. The horizontal line on the left has a ???y???-value of ???-3??? and includes all values of ???x??? in the interval ???x<-2??? (all real numbers ???x??? that are less than ???-2???). For this slice, we write ???-3??? for the function (the constant function whose value is ???-iii???) and ???ten<-2??? for its domain.

The slanted line has a slope of ???5/4??? and a ???y???-intercept of ???-1/2???. To run across how to get the gradient, observe that the points ???(-two,-3)??? and ???(2,2)??? are on this line, then

???y=\frac{5}{4}10-\frac{i}{2}???

For this slice, we write

???f(x)=\frac{5}{4}x-\frac{1}{2}???

for the function and ???-ii\le x\le two??? for its domain.

The horizontal line on the correct has a ???y???-value of ???2??? and includes all values of ???10??? in the interval ???x>two???. For this piece, we write ???2??? for the part and ???x>2??? for its domain.

Putting the iii pieces together, we define this piecewise function as follows:

???f(x) = \begin{cases} -3 & \quad x < -2 \\ \frac{5}{four}x-\frac12 & \quad -ii \leq ten \leq two\\ 2 & \quad ten > two \terminate{cases}???

You might wonder how we decide which piece of this office gets the ???\le??? or ???\ge??? sign and which piece gets the ???<??? or ???>??? sign. The truth is that it doesnâ€™t thing, every bit long as each ???ten??? in the domain of the entire piecewise function is included in the domain of exactly 1 of its pieces – and, of course, that the function for that piece gives the correct value of ???f(x)???. You lot could write it this mode, too:

???f(x)=\begin{cases}-iii & \quad x \leq -2 \\\frac{5}{iv}x-\frac{1}{two} & \quad -2 < x <ii\\two & \quad ten \geq ii\end{cases}???

But information technology could not exist written as

???f(x)=\begin{cases}-3 & \quad ten \leq -2 \\\frac{five}{4}x-\frac{1}{two} & \quad -2 \leq x \leq 2\\ii & \quad 10 \geq 2\end{cases}???

considering here ???-2??? is included in the domains of two different pieces of the role, and so is ???two???.

**Instance**

What is the definition of the piecewise function shown in the graph.

Going from left to right, the first role of the graph is (part of) the parabola ???y=10^two???, which has its vertex at the origin. (To meet that ???y=ten^2??? is the equation of this parabola, note that information technology passes through the point ???(-2,4)??? and that ???4=(-2)^2???.) Thus the role for this slice is ???x^2???, and its domain is ???10<2???.

The second part of the graph is the point ???(ii,8)???, so the function for this slice is ???8??? and its domain is ???ten=2???.

The last part of the graph is part of the line ???y=-x+8???. To see this, weâ€™ll first compute the gradient from the points ???(8,0)??? and ???(two,6)???, both of which are on this line. Then weâ€™ll use the gradient and the point ???(8,0)??? to go the indicate-slope form of the equation of the line (and then use that to go the slope-intercept form). The slope is

???grand =\frac{6-0}{2-8} =\frac{half-dozen}{-half-dozen} =-1???

The point-slope course of the equation of a line is

???y-y_1=m(ten-x_1)???

Using ???(x_1,y_1)=(8,0)??? and the fact that ???m=-1???, nosotros get

???y-0=-ane(x-8)???

???y=-x+8???

Therefore, the function for the final piece is ???-x+eight??? and its domain is ???x>2???.

Putting the 3 pieces together, we define this piecewise office equally follows:

???f(x) = \begin{cases} x^2 & \quad\text{if}\quad x < 2\\ eight & \quad\text{if}\quad x =ii \\ -10+8 & \quad\text{if}\quad ten>2 \finish{cases}???

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## Write a Piecewise Function for the Graph Below

Source: https://www.kristakingmath.com/blog/piecewise-functions-from-their-graphs