Wize High School Algebra I Textbook (Common Core) > Absolute Value Functions

Piecewise Functions

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Piecewise Functions
A piecewise function is a function composed of 2 or more functions with restricted domains. 

The notation used for a piecewise function is displayed below:
f(x)={Piece OnePiece TwoPiece Three...f(x)=\begin{cases}
\text{Piece~One}\\
\text{Piece Two}\\
\text{Piece Three...}
\end{cases}f(x)=⎩⎨⎧​Piece OnePiece TwoPiece Three...​
where each piece of f(x) represents a function and its domain. 

A continuous piecewise function means:
The function is continuous over the entire domain
You do not have to lift your pencil from the paper in order to sketch its graph 
The output values on the boundary points are equivalent
A discontinuous piecewise function means:
The function has discontinuities (holes, gaps, breaks, asymptotes) over the domain
You have to lift your pencil from the paper in order to sketch its graph
The output values on the boundary points are not equivalent 

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Example 
The following piecewise function, f(x), is graphed below:

There are 2 discontinuities: 
x=0x=0x=0
x∈(2,3)x\in(2,3)x∈(2,3)

The domain of the function is:
[−5,0] ∪ [0,2] ∪ [3,10][-5,0]~\cup~[0,2]~\cup~[3,10][−5,0] ∪ [0,2] ∪ [3,10] 

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The function f(x) can be expressed as 3 separate functions with restricted domains. 


Blue Function:
y=x  where  −5≤x≤0y=x~~\text{where}~~-5\leq{x}\leq{0}y=x  where  −5≤x≤0

Green Function:
y=x2+10,≤x≤2y=x^2+1\quad0,\le x\le2y=x2+10,≤x≤2

Purple Function:
y=5,3≤x≤10y=5,\quad3\le x\le10y=5,3≤x≤10


Therefore, the blue, green, and purple function can be expressed as a piecewise function, f(x):

f(x)={x;−5≤x≤0x2+1;0≤x≤25;3≤x≤10f(x) = \begin{cases}x;& -5\leq{x}\leq{0}\\
x^2+1;& 0\leq{x}\leq{2}\\
5;&3\leq{x}\leq{10} \end{cases}f(x)=⎩⎨⎧​x;x2+1;5;​−5≤x≤00≤x≤23≤x≤10​

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Example: Piecewise Functions
The function, f(x),f(x),f(x), is graphed below. 


Find: 
Any discontinuities for f(x).f(x).f(x).
The domain and range for f(x)f(x)f(x)  

Find a piecewise function that describes f(x).f(x).f(x).

Part 1. 

a. There is only one discontinuity at x=3.x=3.x=3.

b. Domain:

   [−5,10]~~~[-5,10]   [−5,10]
   {x∈R∣ −5≤x≤10}~~~\{x\in\mathbb{R}|~-5\leq{x}\leq{10}\}   {x∈R∣ −5≤x≤10}

Range:

   [0,4]~~~[0,4]   [0,4]
   {y∈R∣ 0≤y≤4}~~~\{y\in\mathbb{R}|~0\leq{y}\leq{4}\}   {y∈R∣ 0≤y≤4}

Part 2. 

Blue Function:

y=−12x;   −5≤x≤0y=-\frac{1}{2}x;~~~-5\leq{x}\leq{0}y=−21​x;   −5≤x≤0

Red Function:

y=x;   0<x<3y=x;~~~0<x<3y=x;   0<x<3

Green Function:

y=(x−5)2;   3≤x≤6y=(x-5)^2;~~~3\leq{x}\leq{6}y=(x−5)2;   3≤x≤6

Purple Function:

y=1;   6<x≤10y=1;~~~6<x\le10y=1;   6<x≤10

Therefore, 
f(x)={−12x;−5≤x≤0x;0 <x<3(x−5)2;3≤x≤61;6<x≤10f(x)=
\begin{cases}
-\frac{1}{2}x;&-5\leq{x}\leq{0}\\\\
x;&0\ <{x}<{3}\\\\(x-5)^2;&3\leq{x}\leq6\\\\1;&6<{x}\leq{10}
\end{cases}f(x)=⎩⎨⎧​−21​x;x;(x−5)2;1;​−5≤x≤00 <x<33≤x≤66<x≤10​
Note that there are a few different ways we could have chosen to include the endpoints on some of these domains while still being correct, as the points at x=0x=0x=0 and x=6x=6x=6 will remain the same whichever function is used.

The graph shown below is the graph of the piecewise function f(x)f(x)f(x).
 
Select the following statements that are true about f(x)f(x)f(x).

A) The domain of f(x) f(x)~f(x) is [−4,10][-4,10][−4,10]

B) The domain of f(x) f(x)~f(x) is [−4,0] ∪ [1,4] ∪ [5,10][-4,0]~\cup~[1,4]~\cup~[5,10][−4,0] ∪ [1,4] ∪ [5,10]

C) f(x)f(x)f(x) is discontinuous between (0,1)(0,1)(0,1) and (4,5)(4,5)(4,5)

D)f(x)={−x2;−4<x<0−3x+1;1<x<415x5<x<10f(x)=\begin{cases}
-x^2;&-4<{x}<0\\
-3x+1;&1<x<4\\
\frac{1}{5}x&5<x<10
\end{cases}f(x)=⎩⎨⎧​−x2;−3x+1;51​x​−4<x<01<x<45<x<10​

E) f(x)={−x2;−4≤x≤0−3x+1;1≤x≤415x5≤x≤10f(x)=\begin{cases}
-x^2;&-4\leq{x}\leq{0}\\
-3x+1;&1\leq{x}\leq{4}\\
\frac{1}{5}x&5\leq{x}\leq{10}
\end{cases}f(x)=⎩⎨⎧​−x2;−3x+1;51​x​−4≤x≤01≤x≤45≤x≤10​

I don't know

Practice: Piecewise Functions
Lynn plans on selling hats at a fundraiser. The wholesale hat company charges Lynn $10 a hat for the first 75 hats. After the first 75 hats purchased up to 150 hats, the price of the hat drops by $2.50/hat. After Lynn purchases 150 hats, the price will decrease to $5/hat. 

Determine a function that models this situation. 

A)
f(x)={10x;1≤x≤757.5x+187.5;75<x≤1505x+562.5;x>150f(x)=
\begin{cases}
10x;&1\leq{x}\leq{75}\\
7.5x+187.5;&75<x\leq{150}\\
5x+562.5;&x>150
\end{cases}f(x)=⎩⎨⎧​10x;7.5x+187.5;5x+562.5;​1≤x≤7575<x≤150x>150​

B)
f(x)={10x;1≤x<757.5x+187.5;75≤x<1505x+562.5;x≥150f(x)=
\begin{cases}
10x;&1\leq{x}<{75}\\
7.5x+187.5;&75\leq{x}<{150}\\
5x+562.5;&x\geq150
\end{cases}f(x)=⎩⎨⎧​10x;7.5x+187.5;5x+562.5;​1≤x<7575≤x<150x≥150​

C)
f(x)={10x;0≤x<757.5x+187.5;75<x<1505x+562.5;x>150f(x)=
\begin{cases}
10x;&0\leq{x}<{75}\\
7.5x+187.5;&75<x<{150}\\
5x+562.5;&x>150
\end{cases}f(x)=⎩⎨⎧​10x;7.5x+187.5;5x+562.5;​0≤x<7575<x<150x>150​

I don't know

Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={3x2+4;x<−25x+a;x≥−2f(x)=\begin{cases}
3x^2+4;&x<-2\\
5x+a;&x\geq-2
\end{cases}f(x)={3x2+4;5x+a;​x<−2x≥−2​

Answer

I don't know

Extra Practice

Piecewise Functions

Practice: Piecewise Functions
Chelsea makes wooden boards and sells them to restaurants. She charges $20/board for the first 25 boards. After the first 25 boards purchased up to 50 boards, the price of the board drops by $5.00/board. After Chelsea sells 50 boards to the restaurant, the price will decrease to $12.50/board. 
Determine a function that models this situation. 

Piecewise Functions

Practice: Piecewise Functions
Leonard is purchasing boxes from a wholesaler for his cosmetic company. The wholesaler sells the first 1000 boxes for $2.50/box. The next 5000 boxes are sold at $2.00/box and any number of boxes purchased beyond this are sold for $1.75/box. 
Determine a function that models this situation. 

Piecewise Functions

Practice: Piecewise Functions
The graph shown below is the graph of the piecewise function f(x)f(x)f(x).
Select the following statements that are true about f(x)f(x)f(x).

Piecewise Functions

Practice: Piecewise Functions
The graph shown below is the graph of the piecewise function f(x)f(x)f(x).
Select the following statements that are true about f(x)f(x)f(x).