- [Voiceover] We're given the function f of X
equals three X squared plus eight X minus two,
and we are asked to calculate the following values.
In general, when using function notation,
f of the input, is equal to the output.
So looking at the function values we're asked to determine,
they're giving us the input and our goal
is to determine the corresponding output.
Which means to determine the value of f of zero,
because the input variable is X,
we substitute zero for X here and here.
So f of zero is equal to three times zero squared
plus eight times zero, minus two.
Again notice how we substituted zero for X
in our function f of X.
And now to find the function value
of the corresponding output, we simplify the right side.
So following order of operations
we would simplify the exponents first.
Of course zero squared is zero, so showing our work
this should be three times zero
plus eight times zero minus two.
Next we multiply, well three times zero is zero
and so is eight times zero.
So we have zero plus zero minus two
which equals negative two.
So f of zero is equal to negative two,
which means when the input is zero,
the corresponding output is negative two.
Next we're to determine the value of f of two,
so now we'll substitute two for the input variable X.
So f of two is equal to three times two squared
plus eight times two, minus two.
Now simplifying, we simplify the exponents first.
Two squared is equal to four, so we have
three times four, plus eight times two, minus two.
Next we multiply.
Three times four is 12, plus eight times two is 16,
and adding and subtracting from left to right,
12 plus 16 equals 28, minus two equals 26.
f of two equals 26, which means for the given function,
when the input is two, the output is 26.
Next we have f of negative two.
So we'll substitute negative two for the input variable X.
f of negative two is equal to three times
the square of negative two.
We need the negative two in parentheses
to make sure we have two factors of negative two.
Then we have plus eight times negative two minus two.
Simplifying, we'll simplify the square of negative two,
which is negative two times negative two
which equals positive four, so we have three times four
plus eight times negative two, minus two.
Multiplying three times four is 12
plus eight times negative two is negative 16
and now adding and subtracting from left to right.
12 plus negative 16 is negative four.
So we have negative four minus two
which equals negative six.
f of negative two equals negative six,
which again means, for the given function
when the input is negative two, the output is negative six.
Next we have f of the quantity X plus one.
So we'll follow the same procedure
even though the input is the expression X plus one.
So if we substitute X plus one for X,
f of the quantity X plus one is equal to three times,
X squared will be the quantity X plus one squared
plus eight X will be eight times the quantity X plus one
and then minus two.
And notice how on the right it says
Do not simplify, which means we can stop here,
we do not have to multiply this out and combine like terms.
So f of the quantity X plus one equals three
times the quantity X plus one squared
plus eight times the quantity X plus one minus two.
And then finally we have f of negative X.
Following the same procedure, we'll substitute
or replace the input variable X with negative X.
So f of negative X is equal to three times the square of
negative X, plus eight times negative X, minus two.
Well the square of negative X is equal to
negative X times negative X, which equals
positive X squared, so this would be three X squared,
and then plus eight times negative X
would be negative eight X, so plus negative eight X,
and then minus two.
We don't have any like terms here
but adding a negative is the same as subtracting a positive
so let's write plus negative eight X
as just minus eight X, which is more common.
So f of negative X equals three X squared
minus eight X minus two.
I hope you found this helpful.