Question

Let f(x) = (x^2+3x-4) and g(x) = (x+4)
Find f*g and state the domain.
Find f/g and state the domain.

Help! I need help solving, I think I know the first one just need some advice, but the second one I have no clue!!!!!

Answer 1

First one,
(f*g)(x) = fx*gx= (x^2+3x-4)(x+4)
x(x^2+3x-4)+4(x^2+3x-4)
x^3+3x^2-4x+4x^2+12x-16
x^3+7x^2+8x-16

Answer 2

For f*g , just multiply the two functions: (x^2+3x-4)(x+4)
The first expression can be factored: (x^2+3x-4) = (x+4)(x-1)
f*g = (x+4)(x-1)(x+4) = (x-1)(x+4)^2

You can leave it as factors or multiply them out.

Domain means what are the allowed values of x. Here we see no restrictions on x. x can be any real number. So the domain is -infinity < x < infinity or in interval notation: (-infinity, infinity).

Answer 3

f/g = (x^2+3x-4) / (x+4) = (x+4)(x-1) / (x+4)

Since we cannot divide by zero, x cannot be -4. So the domain must exclude -4
Domain is: (-infinity, -4) union (-4, infinity)

You may be tempted to cancel out the (x+4) in the numerator and denominator and say there is no restriction on the domain but that would be incorrect. When x = -4 it is dividing 0/0 and you can't cancel out the zeros.

But for all x not equal to -4 f/g = (x-1) (very important to remember that x cannot be -4).

Answer 4

So is my first answer in the ballpark?

Answer 5

Your first answer is correct. Just include the domain.

Answer 6

K, how would approach solving the second?

Answer 7

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

Answer 8

Yes and the domain should exclude -4. See my answer above where I have solved the second question.

Answer 9

So it can't be reduced further

Answer 10

You can say either way:
f/g = (x-1) where x is not equal to -4 or the domain is (-infinity, -4) U (-4, infinity)

or f/g = (x^2+3x-4)/(x+4) where x is not equal to -4 or the domain is (-infinity, -4) U (-4, infinity)

I do not know which form your teacher or text book prefers. I will look for similar examples in the book and express it the same way. But both are correct. The first one is nicely simplified.

Answer 11

so second answer would be (x+4)(x-1) / (x+4) and the domain is all real number except x (not equal to) -4

Answer 12

Yes. But if your book prefers the simplified form then the second answer will be:
(x-1) for all real number for x except -4.

If the function f/g were plotted it will look like this (a gap or hole or discontinuity at x = -4):

Answer 13

yes simplified version is the best, so I get the domain part but what is f/g ?

Answer 14

Right so the first one simplifies to x^3+7x^2+8x-16 what does the second one simplify too?

Answer 15

We have been discussing the second solution for quite a while now. The last few replies are all for the second question.

Answer 16

Right I understand that, especially the domain aspect I just wasn't sure other than factoring it to (x+4)(x-1) / (x+4), if that's all i needed to do to find f/g

Answer 17

After factoring you cancel out the (x+4) at the top and the bottom. That leaves with just (x-1) at the top.

f/g(x) = x -1 for all x not equal to -4.

Answer 18

Got it thanks! sorry i little slow, my teacher often gets impatient haha

Answer 19

np. You are welcome.