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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

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}
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=0
  • x∈(2,3)x\in(2,3)

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

Green Function:
y=x2+10,≤x≤2y=x^2+1\quad0,\le x\le2

Purple Function:
y=5,3≤x≤10y=5,\quad3\le x\le10


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}

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Example: Piecewise Functions

The function, f(x),f(x), is graphed below.


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

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

Part 1.

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

b. Domain:

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

Range:

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

Part 2.

Blue Function:

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

Red Function:

y=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}

Purple Function:

y=1;   6<x≤10y=1;~~~6<x\le10

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}
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=0 and x=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).
Select the following statements that are true about f(x)f(x).
Extra Practice
Continuity and Piecewise Functions
What values of aa and bb make the following function continuous?
f(x)={ax2−4;x<−2ax+b;−2≤x≤3ax2+bx+10;x>3f(x)= \begin{cases} ax^2-4;&&x<-2\\\\ ax+b;&&-2\leq{}x\leq3\\\\ ax^2+bx+10;&&x>3 \end{cases}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={x2+10;x<1−x+7a;x≥1f(x)=\begin{cases} x^2+10;&x<1\\ -x+7a;&x\geq1 \end{cases}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={−2x2+7;x<−43x+a;x≥−4f(x)=\begin{cases} -2x^2+7;&x<-4\\ 3x+a;&x\geq-4 \end{cases}
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).
Select the following statements that are true about f(x)f(x).
Piecewise Functions
Practice: Piecewise Functions
The graph shown below is the graph of the piecewise function f(x)f(x).
Select the following statements that are true about f(x)f(x).
Piecewise Functions
Practice: Piecewise Functions
The graph shown below is the graph of the piecewise function f(x)f(x).
Select the following statements that are true about f(x)f(x).
Piecewise Functions
Practice: Piecewise Functions
The graph shown below is the graph of the piecewise function f(x)f(x).
Select the following statements that are true about f(x)f(x).
Piecewise Functions
The graph shown below is the graph of the piecewise function f(x)f(x).
Select the following statements that are true about f(x)f(x).
Piecewise Functions
The following graph is of y=b[[cx]]y=b[[cx]] (Where [[ ]] denotes the greatest integer.) What are b and c?