In this lesson, we will determine
the key characteristics of a piecewise defined function
by analyzing the graph of the function.
Number one, using interval notation,
we're asked to find the domain of the given
piecewise defined function.
The domain is the set of all possible inputs
or x values, and therefore to find the domain
we analyze the graph from left to right.
Notice the left-most point on the graph
is this point here, where x equals negative three,
and because the point is closed at x equals negative three,
negative three is in the domain of the function.
Notice the right-most point on the function
is this point here where x equals five.
Again, that point is closed,
and therefore x equals five is in the domain.
From here, notice from x equals negative three
to x equals positive five, there are no holes
or breaks in the graph, and therefore the domain
is the closed interval from negative three to positive five
which we indicate, using interval notation,
as open square bracket, negative three,
comma, positive five, close square bracket.
Remember, if the endpoints were not included,
we would use parentheses rather than square brackets.
Next, we're asked to determine the range
using interval notation, where the range is the set
of all possible outputs or y values.
To find the range, we analyze the graph vertically.
Notice how the lowest point on the graph
is this point here, where y is equal to negative four,
and the point on the graph here is closed,
and therefore y equals negative four is in the range.
The highest points on the graph are up here
where y equals positive two,
and again, there are no holes or breaks in the graph
from y equals negative four to y equals positive two,
and therefore the range is a closed interval
from negative four to positive two,
which we indicate as open square bracket, negative four,
comma, positive two, close square bracket.
Next, we're asked to determine f of four,
where four is the input or x value.
To determine the function value,
we go to the x-axis and locate positive four,
and in this case go down to the graph and locate the point.
Notice how the point where x equals positive four
is the point four, comma, negative three,
where four is the x value and negative three
is the y value or function value.
F of four is equal to the function value
or y value, negative three.
Next, we're asked to solve f of x equals one.
So here we are given the function value or y value,
and we're asked to find the inputs, or x values.
So we first locate one on the y-axis, which would be here,
and then locate the points on the graph
where y is equal to one.
Notice how there are two of them.
On the left, we have the point negative 2.5, comma, one.
On the right, we have the point 0.5, comma, one.
So you notice how the y value or function value
is positive one when the input or x value
is negative 2.5 or 0.5.
And therefore the solutions are x equals negative 2.5,
comma, 0.5.
Number five, we're asked to find the y-intercept
or vertical intercept of the graph.
This is where the graph touches or crosses the y-axis,
which would be this point here,
where notice how the ordered pair is zero, comma, two.
So the y-intercept is the point given by
the ordered pair, zero, comma, two.
Number six, we're asked to find the x-intercepts
of the graph, which is where the graph
touches or crosses the horizontal or x-axis.
Notice how this occurs in two places.
The graph touches the x-axis at the point
negative three, comma, zero,
and crosses the x-axis at the point one, comma, zero.
So we have two x-intercepts, which we give as ordered pairs,
negative three, comma, zero, and one, comma, zero.
Number seven, we're asked to find the zeroes.
Remember, the zeroes are related to the x-intercepts.
The zeroes are the x values for which
the corresponding function value or y value is zero.
And since one x-intercept is negative three, comma, zero,
we know when x equals negative three,
the y value or function value is zero,
and therefore one zero is x equals negative three.
We also know from the x-intercept of one, zero,
when x is one, the corresponding output
or function value is zero, and therefore
x equals one is also a zero.
So we say the zeroes are x equals negative three,
comma, positive one.
Next, we're asked to find the minimum value of the function.
To find the minimum value, we locate the lowest point
on the graph, which is down here.
Notice how the ordered pair for this low point
is three, comma, negative four.
So the minimum function value is the y value
of negative four, and the x value is the location.
So for number eight, we say the minimum function value
if f of x equals the y value of negative four,
which we can also say occurs at x equals positive three.
Next, we're asked to find the maximum value of the function.
So now we locate the highest point on the graph.
In this case, there are several high points.
All the points along this horizontal segment here
have the maximum y value or function value
of positive two, and therefore we say
the maximum value is f of x equals the y value of two.
Then for number 10, we're asked to determine
if the function is increasing, decreasing or constant
over the given open intervals.
To determine whether the function is increasing,
decreasing or constant, we analyze the graph
from left to right.
If the graph is going uphill from left to right,
then the function is increasing over the interval.
If the graph is going downhill from left to right,
then the function is decreasing over the interval,
and if the graph is horizontal over the interval,
then the function is constant over the interval.
So first we have the open interval from negative three
to positive two, and these are x values.
So we analyze the graph from x equals negative three
to x equals negative two, which is this piece of the graph.
Notice how, from left to right, the graph is going uphill,
and therefore the function is increasing
over this open interval.
Next, we have the open interval from x equals negative two
to x equals zero, which is this piece of the graph.
Notice how the graph is horizontal over this interval,
and therefore the function is constant over this interval.
Next, we have the open interval from x equals zero
to x equals three, which is this piece of the graph.
Again, from x equals zero to x equals three,
notice how the graph is going downhill from left to right,
and therefore the function is decreasing over this interval.
And then finally we have the open interval
from x equals three to x equals five,
which is this right-most piece of the graph.
Notice how over this interval, the graph is going uphill
from left to right, and therefore the function is increasing
over this interval.
I hope you found this helpful.