How to Evaluate a Piecewise Function (example) - Functions Algebra I, Mathematics Video Lecture - Engineering Mathematics

- [Instructor] Consider the
following piecewise function
and we say f(t) is equal to
and they tell us what it's equal to
based on what t is,
so if t is less than or equal to -10,
we use this case.
If t is between -10 and -2,
we use this case.
And if t is greater than or equal to -2,
we use this case.
And then they ask us
what is the value of f(-10)?
So t is going to be equal to -10,
so which case do we use?
So let's see.
If t is less than or equal to -10,
we use this top case, right over here
and t is equal to -10,
that's the one that
we're trying to evaluate.
So we wanna use this case right over here.
So f(-10) is going to be equal to -10,
everywhere we see a t here,
we substitute it with a -10.
- 10 squared minus 5 times,
actually I don't have a denominator there,
I don't know why I wrote it so high.
So it's gonna be -10
squared minus 5 times -10.
So let's see.
- 10 squared, that's positive 100
and then negative,
or subtracting 5 times -10,
this is going to be subtracting -50
or you're going to add 50,
so this is going to be equal to 150.
f(10) is 150,
'cause we used this case up here,
'cause t is -10.
Let's do another one of these examples.
So, here we have
consider the following
piecewise function, alright.
What is the value of h(-3)?
See when h is -3,
which case do we use?
We use this case if our x
is between negative infinity
and zero.
And -3 is in between
negative infinity and zero,
so we're gonna use this
case right over here.
If it was positive three,
we would use this case.
If it was positive 30,
we would use this case.
So we're going to use the first case again
and so for h(-3),
we're gonna take -3 to the third power.
So let's see.
h(-3) is going to be -3 to the third power
which is -27.
And we're done.
That's h(-3).
Because we're using this case,
you could almost ignore
these second two cases
right over here.
Let's do one more example.
This one's a little bit different.
Below is a graph of the step function g(x)
so we can see g(x) right over here.
It starts when x equals -9, it's at 3,
and then it jumps up,
and then it jumps down.
Match each expression with its value.
So g(-3.0001),
so -3.0001,
so that's right over here
and g of that,
we see is equal to 3.
So this is going to be
equal to 3 right over here.
g(3.99999)
3.99999, almost 4,
so let's draw a dotted line right here,
it's gonna be almost 4,
well g(3.99999) is going to be 7.
We see that right over there.
So that is equal to 7.
g(4.00001).
So g(4) is still 7,
but as soon as we go above 4,
we drop down over here,
so g(4.00001) is going to be -3.
I wanna, actually, let's focus
on that a little bit more.
How did I know that?
Well I know that g(4) is 7 and not -3
because we have this dot
is circled in up here
and it's hollow down here.
But as soon as we get
any amount larger than 4,
then the function drops down to this.
So 4.0000, as many,
just slightly above 4,
the value of our function
is going to be -3.
Now let's do g(9).
So g(9), that's when x is 9
and we go down here.
You might be tempted to say it's -3,
but you see, at this
point right over here,
we have an open circle.
So that means that while it's not,
you can't say that the
function is -3 right over there
and there's no other place
where we have a filled-in circle
for x equals 9
so the function g actually
isn't defined at x equals 9.
So I'm gonna put undefined
right over there.