We're asked to graph the piecewise function,
then determine f[0], f[3], and f[6].
Notice how the piecewise function has three pieces,
and notice each piece is a linear function.
Also remember it takes two points to graph a line.
Let's begin by determining a point on the line
y equals negative 2/3 x plus four
for x less than or equal to negative six.
We'll do this using a table of values.
Remember, since our denominator is three here,
if we pick a multiple of three for x,
the y-coordinate will be an integer,
which will be easier to plot.
So, let's plug x equals negative six,
which is in the interval x is less than
or equal to negative six and is a multiple of three.
And now, let's find the corresponding y value.
We have y equals negative 2/3 times x,
which is negative six or negative six over one, plus four.
Simplifying before multiplying,
notice three and negative six
share a common factor of three.
There's one three in three.
There are two threes in six,
but because we have negative six,
negative six simplifies to negative two,
which gives us y equals negative two times negative two
all over one, which is positive four plus four,
which is equal to eight.
Let's go ahead and plot this point on the coordinate plane.
So, again, we have negative six comma eight,
which is here.
Again, it's a closed point here
because x equals negative six
is in the interval x is less than or equal to six
for the function rule y equals negative 2/3 x plus four.
Now, also notice the slope of this line is negative 2/3.
So, if m is equal to negative 2/3,
remember the slope gives us the change of y
over the change of x,
sometimes referred to as the rise over the run.
So, from this point,
we want to find a point on the left of this point,
again, because the interval
is x is less than or equal to negative six.
Notice from this point, if we go down two and right three,
x would be negative three, which is not in the interval
x less than or equal to negative six.
So, let's write the slope
as positive two over negative three.
Using this form of this slope,
we would go up two and left three.
So, if we go up two and left three,
notice how the x-coordinate of this point is in the interval
where x is less than or equal to negative six.
Let's go and sketch the line
to the left of x equals negative six.
Of course, we also could've just found a second point
on the line by selecting an x value
that's a multiple of three less than negative six.
For example, negative nine.
If we substitute x equals negative nine
into the linear equation, we would get y equals 10,
which happened to be this point here
that we just found on the left.
The next function rule is y equals negative four
only when x is greater than negative six
and less than or equal to positive three.
Notice how negative six is not in the interval,
so we will have an open point at x equals negative six.
X equals three is in the interval,
and therefore, we have a closed point at x equals three.
And the graph of yeah equals negative four
is a horizontal line passing through
the y-axis at negative four.
So, to graph this piece,
let's go to negative six on the x-axis,
and then down to negative four for the y-axis,
which would be here, and make an open point.
Again, it's open because x equals negative six
is not in the interval where x is greater than negative six.
And we have a horizontal line to the right,
all the way out to x equals three, which is here.
And we have a closed point at x equals three,
again, because x equals three is in the interval.
And then, finally, we have y equals two x minus three
when x is greater than three.
Three is not in the interval,
and therefore, will have an open point on the line
y equals two x minus three to start this piece.
So, let's plug x equals three for our table,
and substitute three for x in the equation
to find the corresponding y value.
Y is equal to six minus three, which is equal to three.
So, the point three comma three would be an open point
on the line y equals two x minus three.
Let's plot that point.
Three comma three is here.
Make an open point.
And then from here, notice the slope of this line is two,
or two over one.
So, using the slope two over one,
if we go up two and right one, the x value is positive four,
which is in the interval x greater than three.
Now, we can go in and sketch the line on the right.
Another way to graph this piece
would be to find a second point on the line.
For example, when x is equal to four,
notice how y is equal to two times four, which is eight,
minus three, which is five.
And four comma five is the second point that we found
using the slope.
And now, let's find the function values
f[0], f[3], and f[6].
To find the function value f[0], notice x equals zero
is in the interval where x is greater than negative six
and less than equal to positive three.
And therefore, we use the function rule
y equals negative four,
and therefore f[0] is negative four.
And we can verify this graphically.
If we look at the graph, notice when x is equal to zero,
if we go down to the function,
the function value or y value is negative four.
Next, we have f[3].
Notice x equals three, again, is in the interval
where x is greater than negative six
and less than equal to positive three,
and therefore, the y value or function value,
again, is negative four.
If we go back to the graph, notice when x equals three,
the closed point is down here
where the y value or function value is negative four.
And then, finally, we have f[6].
X equals six is in the interval
where x is greater than three,
and therefore, we use the function rule
y equals two x minus three to find f[6].
Substituting six for x, we have two times six minus three,
which is 12 minus three, which is equal to nine.
Going to the graph when x equals six,
we can see the function value is positive nine.
I hope you found this helpful.