Question

Express the function graphed on the axes below as a piecewise function.1086---2

Answer 1

From the graph, we see that this piecewise function will have two expressions, both lines. The first will has the interval:

$x<-5$

Which we can see because it is a line that starts in -5, but not equal to -5 because it is a empty dot, and goes left, to lower values of x.

Similarly, the second part will be in interval:

$x>2$

Because it starts at 2, but not equal to 2, and goes to greater values of x.

To find the expression, we need 2 points. From the graph, we can pick the points (-10,0) and (-5,-5). The equations is:

$y=ax+b$

So, using the points:

$\begin{gathered} -10=a\cdot0+b \\ b=-10 \end{gathered}$

And:

$\begin{gathered} -5=a\cdot(-5)+b=-5a-10 \\ -5+10=-5a \\ 5=-5a \\ a=-1 \end{gathered}$

So, the first expression is

$y=-x-10$

The second expression is alittle bit easier, because it is a constant line at y = -3, so the expression is this:

$y=-3$

Thus, the piecewise function is:

$\begin{gathered} x<-5\colon \\ y=-x-10 \\ x>2\colon \\ y=-3 \end{gathered}$