分段函数

Split the equation into pieces

Before we go into solving absolute value as a piecewise function, let's first look at piecewise functions!

Piecewise functions are functions that have more than one piece to them. They follow different equations depending on what value of $x$. For example, take a look at the following equation:

$$y =\left\{\begin{matrix} x+2\,\,\,\,\text{if}\,\,\,\,x\geq-1\\-4x-2\,\,\,\,\text{if}\,\,\,\,x<-3\end{matrix}\right.$$

If you think about this function as a machine, you would have to sort your $x$'s before you dropped them in. If $x$ is less than $-3$, you have to drop it in the bucket that goes to the equation $-4x-2$. If it's greater than or equal to $-1$, you have to drop it in the bucket that goes to $x + 2$. A graph of this function looks like this:

These equations don't form one straight, continuous line, but they're all part of the same function.

Absolute value functions work the same way on a graph. In fact, their graphs are always the shape of a V. Take a look at the graph below:

This graph is for the function $f(x) = |2x + 6|$. Notice that there seem to be two lines. One has a slope that climbs up, another slides down. We can find the equations for these two lines by going through the steps to solve absolute value functions.

First, separate the two options out. We can say that $f(x) = |2x + 6|$ is the same as

$$f(x) = 2x + 6\,\,\,\,\text{if}\,\,\,\,2x + 6 \ge 0\\ \text{and}\\ f(x) = -(2x + 6)\,\,\,\,\text{if}\,\,\,\,2x + 6 < 0$$

When the value of $2x + 6$ dips below $0$, the direction of the line changes. We can formalize those by just putting them into one equation:

$$y=\left\{\begin{matrix} 2x+6\,\,\,\,\text{if}\,\,\,\,2x+6\geq0\\-(2x+6)\,\,\,\,\text{if}\,\,\,\,2x+6<0\end{matrix}\right.$$

...and then simplifying the inequalities!

$$y=\left\{\begin{matrix} 2x+6\,\,\,\,\text{if}\,\,\,\,x\geq-3\\-2x-6\,\,\,\,\text{if}\,\,\,\,x<-3\end{matrix}\right.$$

Then, you can find the graph by just plugging the $x$ values into the equation that they go to! These easy steps will allow you to interpret absolute functions as piecewise.

Try it!

Take a look at the following function: $$f(x) = |-x + 5|$$

Right, when the segment of the equation that's within the absolute value brackets is equal to a number greater than or equal to $0$, everything works normally.

When the value of $-x + 5$ is less than $0$, the function will spit out the positive equivalent of that number. We can say that the equation it's actually following is:

$$-1(-x + 5)\\ -(-x + 5)\\ x - 5$$

Awesome! We know that the function follows $-x + 5$ if $-x + 5 \ge 0$ and $x - 5$ if $-x + 5 < 0$. Write out the piecewise function that means the same thing (don't forget to simplify the inequalities!).

$$\begin{align} f(x)&=|-x+5|\\\ \\ f(x)&=\left\{\begin{matrix} -x+5\,\,\,\,\text{if}\,\,\,\,-x+5\geq0\\x-5\,\,\,\,\text{if}\,\,\,\,-x+5<0\end{matrix}\right.\\\ \\ f(x)&=\left\{\begin{matrix} -x+5\,\,\,\,\text{if}\,\,\,\,x\ge 5\\x-5\,\,\,\,\text{if}\,\,\,\,x<5\end{matrix}\right. \end{align}$$

Nice! Now try graphing those equations on a piece of paper, making sure to stop the lines when they get outside of their limit in the piecewise function.

(Video) Writing an Absolute Value as a Piecewise Function

by Daniel Kopsas

This video by Daniel Kposas explains the connection between absolute values and piecewise functions.

$\large{\text{Summary}}$

We can graph absolute value functions by translating them into piecewise functions. Piecewise functions are functions with more than one piece. Each piece follows different equations depending the value of $x$. If you remember how to solve absolute value equations, they should look pretty similar.

Let's break down an absolute value.

$$y=|x+2|$$

First, we set up the boundaries for the domain. These boundaries tell us which part of the function to use based on the $x$-values we are using. To find the boundaries, we use the expression inside the absolute value sign. Remembering that absolute value gives distance from $0$, we use $0$ as the divider between the boundaries.

$$\begin{align} x+2 \ge 0 \\ x+2<0 \end{align}$$

Second, we set up the rest of the function. Assign the expression inside the absolute value signs, $x+2$, to the first boundary. And the opposite of the expression, $-(x+2)$ to the second boundary.

$$\begin{align} y&=\left\{\begin{matrix} x+2\,\,\,\,\text{if}\,\,\,\,x+2\geq0\\-(x+2)\,\,\,\,\text{if}\,\,\,\,x+2<0\end{matrix}\right.\\\ \end{align}$$

Now we can isolate $x$ in each inequality. This will give us the $x$-value for the vertex.

$$\begin{align} x+2 &\geq 0\\ x&\geq -2\\ \\ x+2 &< 0\\ x &< -2 \end{align}$$

We can then apply these to our piecewise function.

$$y=\left\{ \begin{matrix} x+2\,\,\,\,\text{if}\,\,\,\,x\geq-2\\ -x-2\,\,\,\,\text{if}\,\,\,\,x<-2 \end{matrix}\right.$$

For more practice, and to make these functions a little bit more complex, try word problems with absolute value inequalities!

Interpret Absolute Value Functions as Piecewise

When interpreting an absolute value function as a piecewise function (which could be helpful for graphing), it is important to remember that the expression within the absolute value could be negative OR non-negative. In other words, the expression within the absolute value could be $<0$ or $\ge 0.$ Use those two cases to split up the absolute value accordingly, and then simplify!