Video Transcript
Give the piecewise definition of
the function 𝑓 whose graph is shown.
We’re told that this graph
represents the graph of a piecewise-defined function. And this makes a lot of sense. We see that it’s made up of three
different parts. We have a linear function over here
given by a single straight line and a other linear function here given by another
straight line. But then we have something really
strange here. We have a single dot at this
point. And we’ll consider what that means
for our piecewise definition in a moment.
For now, we’re going to begin by
finding the equation of our two straight lines. We use the formula 𝑦 minus 𝑦 one
equals 𝑚 times 𝑥 minus 𝑥 one, where 𝑚 is the slope of the graph and 𝑥 one, 𝑦
one is a single point it passes through. Then the slope is given by change
in 𝑦 divided by change in 𝑥, which is 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥
one. And so let’s begin by finding the
slope of our first line, we can choose any two points on this line. Let’s choose the points with
coordinates negative three, six and one, two. Then change in 𝑦 divided by change
in 𝑥 is six minus two over negative three minus one. Of course, we could write two minus
six over one minus negative three and get the same result.
This gives us four divided by
negative four, which is negative one. Then substituting everything we
know about our first straight line into the formula for a straight line, and we get
𝑦 minus six equals negative one times 𝑥 minus negative three. Distributing the parentheses on the
right-hand side, and this simplifies to negative 𝑥 minus three. Finally, we add six to both sides,
and we find 𝑦 is equal to negative 𝑥 plus three or three minus 𝑥. And so for values of 𝑥 strictly
less than two, we can use the equation 𝑦 equals three minus 𝑥 to draw its
graph.
Next, we choose two points on our
second line. Let’s choose the points with
coordinates four, four and six, five. Change in 𝑦 divided by change in
𝑥 here is five minus four over six minus four, which is equal to one-half. So the slope of our second line is
one-half. Substituting then 𝑚 equals
one-half and 𝑥 one, 𝑦 one equals four, four into our formula for a straight line,
and we get 𝑦 minus four equals a half times 𝑥 minus four. And that right-hand side simplifies
to 𝑥 over two minus two. Then we add four to both sides. And we see that our second line has
the equation 𝑦 equals 𝑥 over two plus two. Now, this time, that’s for values
of 𝑥 strictly greater than two. So we now have the equations of our
two straight lines. These are three minus 𝑥 if 𝑥 is
less than two and 𝑥 over two plus two if 𝑥 is greater than two.
We’re not finished though; there
was a third subfunction that we’re interested in. And this subfunction is represented
graphically by a single point. This point has coordinates two,
two. In other words, if 𝑥 is exactly
equal to two, the function yields an output of two. And so that is our third
subfunction. Noting that we can alternatively
write the domain on our third line as two is less than 𝑥, we now have the piecewise
definition of our function. It’s 𝑓 of 𝑥 is equal to three
minus 𝑥 if 𝑥 is less than two, two of 𝑥 is equal to two, and 𝑥 over two plus two
if two is less than 𝑥.
