Question

PLEASE HELP!!!

let f(x)=(x^2+3x-4) and g(x)=(x+4)

a. find f*g and state the domain
b. find f/g and state the domain

Answer 1

Well, do it. What is f*g?

Answer 2

Thats just it im not sure

Answer 3

I do have to wonder why you have been given such a problem statement with NO introduction to the notation. That just makes no sense.

f*g = f(x) * g(x)

Now, can you produce f*g?

Answer 4

alright but how do i do f(x) when im not given an x to begin with

Answer 5

f/g = f(x) / g(x) = (x^2+3x-4) / (x+4)

Answer 6

You are given a definition of f(x). You do NOT need a specific value for 'x'. The definition of f(x) is a relation for ALL values of x. You are being asked to consider and algebraic manipulation, not a numerical result.

Answer 7

alright

Answer 8

Your turn. What is f*g?

Answer 9

x^3+7x^2+8x-16??

Answer 10

Very good.

It may have been sufficient to write f*g = (x^2+3x-4)*(x+4).

Okay, that is your, basic, run-of-the-mill polynomial Relation. What is the Domain of such a relation?

Answer 11

wouldnt that be all real numbers?

Answer 12

Excellent. All Real Numbers.

How about f/g?

Answer 13

x-1??

Answer 14

No, that's no good. It may be closely related to h(x) = x-1, but it is not quite the same thing.

Please do the following exercise:

If h(x) = x - 1
And if r(x) = (x^2+3x-4) / (x+4)

Find h(-4) and r(-4).

Let's see if we learning anything that is not obvious at the moment.

Answer 15

h(-4)=-5 and 0
right?

Answer 16

Nope. Try r(-4) again. Do the numerator and denominator separately. Perhaps you will see the problem.

Answer 17

That was pretty good, though. We did establish that h(x) and r(x) are NOT the same thing. You just told be -5 and 0. Not the same.

Answer 18

i just did that and got that same thing

Answer 19

it doesnt seem to work

Answer 20

You should not get zero. You are not understanding what you are seeing.

Let's learn something about zero (0).

12/4 = 3 can be expressed as 3*4 = 12
6/3 = 2 can be expressed as 2*3 = 6
14/2 = 7 can be expressed as 2*7 = 14
-12/2 = -6 can be expressed as -6(2) = -12

Make sense so far? Bear with me. It will be worth it when we get there!

Answer 21

okay

Answer 22

Is there another expression for 0/0?

Answer 23

0*0??

Answer 24

If 0/0 = 0, we should be able to express it as 0*0 = 0, but isn't it also the case that we can express it as 0*5 = 0 or 0*12 = 0 or 0*(-6) = 0? It doesn't really matter what that other value is. If the first is zero, there is no significance to the second value. In the second value, 0 just isn't important.

Any sense at all, yet?

Answer 25

yes

Answer 26

Here's the kicker. Is is very important that you get this one stuck in your head.

Zero in the denominator doesn't mean anything. It just doesn't matter what's int eh numerator.

0/0 is NOT 1. It's also not "infinity". It's also not 5 and it's not the square root of 2.

When you evaluated r(-4), you should get zero for both numerator and denominator separately. I believe you did that, but then you wanted to interpret that as 0/0 = 0 and that is no good.

What we have is this:

f/g is a rational function with x = -4 NOT in the Domain. You absolutely may not use x = -4 in f/g.

When you wrote f/g = x-1 you were ALMOST correct. This is correct for EVERY Real Number EXCEPT x = -4. As long as you avoid x = -4, then f/g = x-1.

Soaking it all up? It is glorious to behold!

Answer 27

so then the domain is \[x \neq 4\] and f/g=x-1??

Answer 28

all real numbers except -4