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Wize High School Grade 12 Pre-Calculus Textbook > Functions (Basics)

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).

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.

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}

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).