To write the for the given graph, we need to find the equation of the line representing each piece and then restrict the accordingly.
The First Piece
Let's begin by finding the and of the first piece of the function.
Notice that the graph is a part of a . We know that horizontal lines have a slope of
0. The
y-intercept of our line is
4. We can write the equation for this piece in .
y=0x+4⇔y=4
The Other Pieces
Now let's take a look at the remaining pieces. Again, we can see that all of these pieces are horizontal. Therefore, we find their slope-intercept form as we did for the first one. To help us find the y-intercepts, we can extend the lines to the y-axis.
Let's write the equations for these pieces in slope-intercept form.
Second piece:y=0x+3⇔y=3Third piece:y=0x+2⇔y=2Fourth piece:y=0x+1⇔y=1
Combining the Pieces
We can add the equations of these lines to the piecewise .
f(x)=⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧4321
Finally, we need to determine the domain for each equation. The
jumps
occur at
x=1, x=2 and
x=3. These will be where the domains are divided. We cannot have an overlap in our domains so we need to take notice of where the closed and open circles are located.
The open circle at the beginning of the first piece tells us that its domain does not include
0. The closed circle at the end of the first piece tells us that its domain includes
1. Similarly, we can find the domains of the remaining pieces.
f(x)=⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧4,3,2,1,if 0<x≤1if 1<x≤2if 2<x≤3if 3<x≤4