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# Graphs of nonlinear piecewise functions

Video transcript

Select the piecewise function
whose graph is shown below. Or I guess we should
say to the right. I copied and pasted it
so it's on the right now. So we have this piecewise
continuous function. So it's not defined for x
being negative 2 or lower. But then starting at x
greater than negative 2, it starts being defined. It's continuous all
the way until we get to the point x equals 2 and
then we have a discontinuity. And then it starts getting
it defined again down here. And then it is continuous for
a little while all the way. And then when x is greater than
6, it's once again undefined. So let's think about
which of these functions describe this one over here. So this one looks like a
radical function shifted. So square root of x
would look like this. Let me do it in a color
that you could see. Square root of x
would look like this. So the square root
of x looks like this. And this just looks
like square root of x shifted 2 to the left. So this looks like
square root of x plus 2. This one right over here looks
like square root of x plus 2. And you could verify that. When x is negative 2, negative
2 plus 2, square root of that is going to be 0. And it's not defined
there, but we see that if we
were to continue it would have been defined there. Let's try some other points. When x is negative 1,
negative 1 plus 2 is 1. Principal root of 1 is 1. Let's try 2. When x is equal to
2, 2 plus 2 is 4. Principal root of
4 is positive 2. So this just looks like
a pretty good candidate. So it looks like our function. So let's see, it looks like
our function would be-- and I'm not going
to call it anything because it could
be p, h, g, or f. But our function, if I
were to write it out, it looks like over
this first interval. So it looks like it's the square
root of x plus 2 for negative 2 being less than x-- it's
not defined at negative 2, but as long as x is greater than
negative 2 and x is less than or equal to 2. And x is less than
or equal to 2. So that's this part
of the function. And then it jumps down. Now, this looks like
x to the third. x to the third looks
something like this. So x to the third power
looks something like that. So let's see, negative 2 to
the third power is negative 8. So it looks something like that. That's what x to the
third looks like. So 2 to the third power is 8. So x to the third looks
something like that. This looks like x to the
third shifted over 4. So I'm guessing that this is
x minus 4 to the third power. But we can verify that. When x is equal to 4, 4
minus 4 to the third power, we do indeed get the value
of this expression being equal to 0. When x is 6, 6 minus 4 is 2 to
the third power is indeed 8. When x is 2, 2 minus 4 is
negative 2 to the third power is indeed negative 8. So this over this
interval, it is x minus 4 to the third power. So we could say so this part of
the function right over here, we could say is x minus 4
to the third power for 4x being greater than 2, or we
could say 2 is less than x. And it's defined all the
way to x being equal to 6, but not being greater than. So x is less than or equal to 6. So which of these choices
are what we just put in? So the square root of x plus 2--
well, that's not-- let's see. Square root of x
plus 2 for negative 2 is less than x is less
than or equal to 2. X minus 4 to the third
for 2 is less than x, which is less than
or equal to 6. So I would go with that one.