We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

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Piecewise-Defined Functions

A piecewise function exists when a function is defined by two or more different functions throughout its domain. The first step in evaluating a piecewise function is to determine which function definition applies depending on the value of x that is being input. Once that has been determined, we evaluate the function as usual by substituting in the given value of x.

Example:

Given $f(x) =
\left\{
\begin{array}{lr}
-x&\text{if}~x<0\\
x^2+1&\text{if}~x\ge0
\end{array}
\right\}$,find $f(-1)$, $f(0)$ and $ f(1) $

$piecewise function$

Solution: First, we want to evaluate $f$ at $x=-1$, so $f(x)=-x$ is the equation to use

(since it applies whenever $x<0$).

$ \displaylines{
f(x) &=& -x \\
f(-1) &=& -(-1) \\
f(-1) &=& 1} $

Next, we want to evaluate $f$ at $x=0$, so $f(x)=x^2+1$ is the equation to use.
(Since it applies whenever $x\geq 0$.)

$\displaylines{f(x) &=& x^2+1 \\ f(0) &=& 0^2+1 \\ f(0) &=& 1} $

Finally, we evaluate $f$ at $x=1$, so $f(x)=x^2+1$ is the equation to use. (Since it applies whenever $x\geq 0$.)

$\displaylines{f(x) &=& x^2+1 \\ f(1) &=& 1^2+1 \\f(1) &=& 2} $

Absolute Value Functions

An absolute value function can be rewritten as a piecewise function. Absolute value is the distance from a number 'x' to 0 on the real number line. Therefore, when the value of a function is equal to zero or is positive, taking its absolute value doesn't change it; however, if the value is negative, taking absolute value changes the sign. Therefore, the definition of the function changes depending on whether or not x ≥ 0.

$y = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} $

Example: Write $y = |x-2|$ as a piecewise function and evaluate $f(0)$, $f(2)$, and $f(5)$.

Solution:

$ f(x) = \begin{cases} x-2, & \text{if } x \geq 0 ~or~x\geq 2 \\ -(x-2), & \text{if } x < 0~or~x<2 \end{cases} $

First, to find $f(0)$, we use $f(x) = -(x-2)$, since this applies when $x<2$:

$ \displaylines{f(x) &=& -(x-2) \\ f(0) &=& -(0-2) \\ f(0) &=& 2} $

Next, to find $f(2)$, we use $f(x) = x-2$, since this applies when $x \geq 2$:

$ \displaylines{f(x) &=& x-2 \\ f(2) &=& 2-2 \\ f(2) &=& 0} $

Finally, to find $f(5)$, we use $f(x) = x-2$, since this again applies when $x \geq 2$:

$ \displaylines{f(x) &=& x-2 \\ f(5) &=& 5-2 \\ f(5) &=& 3} $

Example 1:

Example 2:

Example 3: