Video Transcript
Give the piecewise definition of
the function ℎ whose graph is shown.
We’re told that the graph we’re
given is a piecewise-defined function. And we know that a
piecewise-defined function is made up of multiple subfunctions. In fact, by looking at the graph of
this function, we might notice that there are going to be two subfunctions. These are also going to be linear
since the graph of each subfunction is a straight line. And so this means that if we can
calculate the slope 𝑚 and find one point which each line passes through, we can use
the equation 𝑦 minus 𝑦 one equals 𝑚 times 𝑥 minus 𝑥 one to find the equation of
each line.
Let’s begin with the first part of
this subfunction. We notice that this subfunction is
defined up to and including 𝑥-values of two. So that will give us a hint as to
what its domain is. Then we could use the formula for
slope 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one to find the slope of
this line. Alternatively, we can use the
triangle method. Choosing a point on the line, in
this case, the 𝑦-intercept, and then moving exactly one unit to the right, we see
we have to move one unit down to get back to our point on the line. That means the slope of this line
must be negative one. It also passes through the point
zero, three. Remember, this is the 𝑦-intercept
of the line.
And so substituting everything we
know about this first function into our equation for a straight line, we get 𝑦
minus three equals negative one times 𝑥 minus zero. Distributing the parentheses on the
right-hand side, and we get negative 𝑥. And then we’re going to make 𝑦 the
subject by adding three to both sides. Remember, 𝑦 is the output. So it’s going to be ℎ of 𝑥
essentially. And so the first line is defined by
the equation 𝑦 equals three minus 𝑥.
Let’s repeat this process with the
second line. Now we always need to be a little
bit careful using the triangle method for fractional slope values. In this case, when we pick a point
on the line, move one unit to the right, we then have to move a half a unit up to
get back to our point on the line, meaning that the slope of our second line is
one-half. To convince ourselves that this is
true, we could choose the two points given on the line, which have coordinates four,
two and six, three, respectively. Then 𝑦 two minus 𝑦 one over 𝑥
two minus 𝑥 one, which is change in 𝑦 over change in 𝑥, is three minus two over
six minus four, which is one-half as we saw.
Then let’s pick this point. We know our line passes through the
point with coordinates two, one. And so the equation of our line is
𝑦 minus one equals a half times 𝑥 minus two. Then when we distribute the
parentheses on the right-hand side, we get that one-half times 𝑥 minus two is the
same as one-half 𝑥 or 𝑥 over two minus one. We can then finally add one to both
sides, eliminating that negative one. And so the second line has the
equation 𝑦 equals 𝑥 over two. Now that we have the equations that
represent our subfunctions, we’re going to pop this back together using piecewise
definition.
ℎ is given by three minus 𝑥 for
values of 𝑥 less than two. And 𝑥 over two of 𝑥 is greater
than or equal to two, which of course is the same as writing two is less than or
equal to 𝑥. Note, of course, that the function
could have been defined at the point 𝑥 equals two by either subfunction. It’s generally convention that we
choose the second function to define that point, although it would have been just as
correct to write three minus 𝑥 if 𝑥 is less than or equal to two and 𝑥 over two
if 𝑥 is greater than two. The piecewise definition of ℎ is
three minus 𝑥 if 𝑥 is less than two and 𝑥 over two if two is less than or equal
to 𝑥.
