Introduction

A piecewise function is a function that is defined differently in different parts of the function's domain.

For example, consider the piecewise function f(x), defined as follows: f(x) = \begin{cases} {\color{red}{-2x}} \,, \quad & x\leq 0\\ {\color{blue}{\dfrac{1}{2}x}}\,, \quad & x> 0 \end{cases} A plot of the function is given below:

The function has two distinct branches, one for \color{red}{x\leq 0} and one for \color{blue}{x>0}. The function is a piecewise function because it has different definitions depending on where we are in the domain.

Suppose that we want to calculate f(-1). For this, we'd use the left branch x \leq 0 because -1 \leq 0. On this branch, the function definition is f(x) = -2x, so we have

f(-1) = -2(-1) = 2.

Similarly, if we want to calculate f(1) , we'd use the right branch x > 0 because 1 > 0. On this branch, the function definition is f(x) = \dfrac{1}{2}x, so we have

f(1) = \dfrac 1 2 \cdot 1 = \dfrac 1 2.

FLAG
Example: Evaluating Piecewise Functions

For the function g(x) defined on the set of integers, find g(-12) and g(3), where g(x) is the following piecewise function: g(x) = \begin{cases} 2x \,, \quad & x < 1\\ 3x \,, \quad & x \geq 1 \end{cases}

EXPLANATION

  • First, we compute g(-12). Since -12 < 1, we use the first branch: \begin{align*} g(x) &= 2x \\[5pt] g(-12) &= 2(-12) \\[5pt] &= -24 \end{align*}

  • Next, we compute g(3). Since 3 \geq 1, we use the second branch: \begin{align*} g(x) &= 3x \\[5pt] g(3) &= 3(3) \\[5pt] &= 9 \end{align*}

FLAG
Practice: Evaluating Piecewise Functions

Given the following piecewise function $f(x),$ what is $f(3)?$ \[ f(x) = \begin{cases} \dfrac 1 4 x\,, \quad & x\leq 3\\ 5\,, \quad & x\gt 3 \end{cases} \]

a
 $5$
b
 $\dfrac 3 4 $
c
 $3$
d
 $\dfrac 1 4 $
e
 Undefined
Practice: Evaluating Piecewise Functions

Given the following piecewise function $f(x),$ what is $f(-2)?$ \[ f(x) = \begin{cases} 2x , \quad & x< 0\\ x+3, \quad & x\geq 0\\ \end{cases} \]

a
 $0$
b
 $-5$
c
 $-4$
d
 $4$
e
 $5$
Example: Graphing a Piecewise Step Function

Given that f(x) = \begin{cases} 3, \quad &x< -3\\ 1, \quad &x\geq -3 \end{cases} what is the graph of y=f(x)?

EXPLANATION

We will graph each branch separately.

  • On the interval (-\infty, -3), we have f(x) = 3, so the graph is a horizontal line that ends at (-3,3), excluding this point.

  • On the interval [-3, \infty) , we have f(x) = 1, so the graph is a horizontal line that begins at (-3,1), including this point.

Therefore, the graph of the given function is as follows:

FLAG
Practice: Graphing a Piecewise Step Function

Given that \[ f(x) = \begin{cases} -1\,,\quad &x< 0\\ 2\,, \quad &x\geq 0, \end{cases} \] which of the following is the graph of $y=f(x)?$

a
b
c
d
e
Practice: Graphing a Piecewise Step Function

Given that \[ f(x) = \begin{cases} 1, \quad &x \leq -2\\[3pt] -1, \quad &-2 < x \leq 2\\[3pt] 2, \quad &x \gt 2, \end{cases} \] which of the following is the graph of $y = f(x)?$

a
b
c
d
e
Example: Graphing a Piecewise Function

If f is the piecewise function f(x) = \begin{cases} -1, \quad &x \leq 0\\ -x+2, \quad &x > 0 \end{cases} what is the graph of y=f(x)?

EXPLANATION

We will graph each branch separately.

  • On the interval (-\infty, 0], we have f(x) = -1 , so the graph is a horizontal line that ends at (0,-1), including this point.

  • On the interval (0, \infty) , we have f(x) = - x+2 , so the graph is a straight line of slope -1 that starts from (0,2), not including this point.

Therefore, the graph of the given function is as follows:

FLAG
Practice: Graphing a Piecewise Function

What is the definition of the function $y = f(x)$ shown above?

a
 $ f(x) = \begin{cases} -1, \quad & x \leq 0\\[0pt] -x+3, \quad & x > 0 \end{cases} $
b
 $ f(x) = \begin{cases} 1, \quad & x \leq 0\\[0pt] x-1, \quad & x > 0 \end{cases} $
c
 $ f(x) = \begin{cases} -2, \quad & x \leq 0\\[0pt] x+1, \quad & x > 0 \end{cases} $
d
 $ f(x) = \begin{cases} 2, \quad & x \leq 0\\[0pt] -x+1, \quad & x > 0 \end{cases} $
e
 $ f(x) = \begin{cases} 2, \quad & x \leq 0\\[0pt] x+1, \quad & x > 0 \end{cases} $
Practice: Graphing a Piecewise Function

What is the definition of the function $y=f(x)$ shown above?

a
 $ f(x) = \begin{cases} 2x+1, \quad & x\leq 1\\ 2-2x, \quad & x\gt 1 \end{cases} $
b
 $ f(x) = \begin{cases} 1-x, \quad & x\leq 1\\ x-1, \quad & x\gt 1 \end{cases} $
c
 $ f(x) = \begin{cases} 2x+1, \quad & x\leq 1\\ -2-2x, \quad & x\gt 1 \end{cases} $
d
 The graph is not a function
e
 $ f(x) = \begin{cases} 2x-1, \quad & x\leq 1\\ 1-2x, \quad & x\gt 1 \end{cases} $
Example: The Domain of a Piecewise Function

What is the domain of the function shown below?

EXPLANATION

The function is defined for values between -4 and 1.

  • Since there is an open circle at x=-4, this point is not included in the domain.

  • Since there is a closed circle at x=-1, this point is included in the domain.

  • Since there is an open circle at x=1, this point is not included in the domain.

So, the function f(x) above is defined for values between x=-4 and x=1, not including x=-4 and x=1 but including x=-1.

Therefore, the domain is x\in (-4,1)

Note that the function is defined at the point x=-1, even though the function jumps at that point. The function value at x=-1 is 2.

FLAG
Practice: The Domain of a Piecewise Function

What is the domain of the function $f(x),$ shown above?

a
 $[-1,5)$
b
 $[-1,3)\cup (3, 5]$
c
 $(-1,5]$
d
 $(-1,5)$
e
 $[-1,5]$
Practice: The Domain of a Piecewise Function

What is the domain of the function shown above?

a
 $(-\infty,-1)\cup(-1 ,4]$
b
 $(-\infty,-1)\cup (-1 ,\infty )$
c
 $ (-\infty,-1)\cup(-1 ,4)$
d
 $(-4, -1)\cup (-1,4]$
e
 $(-\infty,4]$
Flag Content
Did you notice an error, or do you simply believe that something could be improved? Please explain below.
SUBMIT
CANCEL