To write the piecewise function for the given graph, we need to find the equation of the line representing each piece and then restrict the domain accordingly.

The First Piece

Let's begin by finding the slope and $y -$ intercept of the first piece of the function.

Notice that the graph is a part of a horizontal line. 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 slope-intercept form.

y = 0 x + 4 \Leftrightarrow 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 = 0 x + 3 \Leftrightarrow y = 3 Third piece : y = 0 x + 2 \Leftrightarrow y = 2 Fourth piece : y = 0 x + 1 \Leftrightarrow y = 1

Combining the Pieces

We can add the equations of these lines to the piecewise function notation.

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 \leq 1 if 1 < x \leq 2 if 2 < x \leq 3 if 3 < x \leq 4

Exercise