We are given f of x and g of x
and asked to find f times g of x
and f times g of negative two.
Number one,
f times g of x
is equal to f of x times g of x.
Because f of x is equal to two x minus three,
and g of x equals x squared minus three x plus two,
f of x times g of x
is the quantity two x minus three times
the quantity x squared minus three x plus two.
Now we need to find the product to do this.
We distribute the two x which gives us three products.
Then we distribute to negative three
which gives us three more products.
Two x times x squared equals two x cubed.
Two x times negative three x equals negative six x squared,
giving us minus six x squared.
And two x times positive two is equal to
positive four x giving us plus four x.
And now we distribute negative three.
Negative three times x squared is negative three x squared
giving us minus three x squared.
Negative three times negative three x is positive nine x
giving us plus nine x.
And finally we have negative three times positive two,
which equals negative six, giving us minus six.
Now we combine like terms.
We have two x squared terms,
and we have two x terms.
We have two x cubed and then negative six x squared
minus three x squared is negative nine x squared,
giving us minus nine x squared.
And four x plus nine x is 13x, giving us plus 13x
and we still have minus six.
Now we know f times g of x equals two x cubed
minus nine x squared
plus 13 x
minus six.
For number two, we're asked to find
f times g of negative two.
Because we just found the function,
f times g of x, one way to find f times g of negative two
is to substitute negative two for x into f times g of x.
The second method defined f times g of negative two
is to determine f of negative two times g of negative two.
In this video we will show both methods.
For the first method, though,
let's use f times g of x that we just found.
So f times g of negative two is equal to
substituting negative two for x and f times g
gives us two times the cube of negative two
minus nine times the square of negative two,
plus 13 times negative two minus six.
We begin by simplifying the exponents.
The cube of negative two equals negative eight.
This gives us two times negative eight
minus nine times the square of negative two
is positive four.
The next step is to multiply.
Two times negative eight equals negative 16,
then we have minus nine times four which gives us minus 36.
Then we have plus 13 times negative two
is equal to negative 26.
And we have minus six.
And now we add, subtract from left to right.
Negative 16 minus 36 equals negative 52,
negative 52 plus negative 26 is equal to negative 78,
negative 78 minus six equals negative 84.
F times g of negative two equals negative 84.
But let's also determine f times g of negative two
by determining f of negative two times g of negative two.
Let's do this on the next slide.
Again, f times g of negative two, using the definition,
is equal to f of negative two times g of negative two.
To find f of negative two, we substitute negative two
for x into f of x, which gives us
two times negative two minus three.
And we have times g of negative two,
which is equal to the square of negative two
minus three times negative two plus two.
Now we simplify inside the parentheses, then multiply.
Two times negative two is equal to negative four,
giving us negative four minus three,
times the quantity the square of negative two
is positive four.
Negative four minus three is negative seven.
Here we have minus three times negative two,
which is minus negative six,
which is equivalent to plus six,
giving us four plus six plus two.
This gives us negative seven times four plus six plus two
is equal to 12.
And a negative seven times 12 is negative 84,
of course giving us the same result.
So you may notice, if we hadn't already found
f times g of x in number one,
it would probably be faster to find
f times g of negative two by using this method here.
I hope you found this helpful.