Composite Functions Homework | f/g(x)

[Corkery]

Homework Statement

Given f(x) = 3/x-7 and g(x) = x^2 + 5x find each funstion below (f/g) (x)

Homework Equations

The Attempt at a Solution

Okay so I think I have most of these steps but I think the only thing I'm missing is the factoring, which i sort of forgot over the summer. take it easy on me i just started school.

a. 3/x-7 / x^2 + 5x
b. = 3/x-7 x 1/x^2 + 5x
c. = 3/x^3 + 5x^2 - 7x^2 - 35x
d. this is where i get confused but ill take a stab at it. so combine like terms on the denominator. x^3 - 2x^2 - 35x and after this i think you factor it...that is if i did it right. And if i did this right I got (1x-7) (1x+5) = 7,-5

 

[Mindscrape]

Do you have to simplify your answer? I think

$$\frac{3}{x^3 -2x^2 -35x}$$

is good enough. Although, if you want to factor it then factor out an x and figure out the quadratic factorization, which you sort of did. (x-7)(x+5) does not equal 7 and -5 though.

$$\frac{f(x)}{g(x)} = \frac{3}{x(x-7)(x+5)}$$

What would happen if x was 7 or -5?

 

[Corkery]

honestly i get what you just showed and explained to me but am clueless to what to do next? any hints please.

 

[Mindscrape]

Maybe I am missing something in the problem description, but you would be done if you wrote down my last equation in terms of finding f/g (x). You still didn't answer my question though. Let's rename f/g (x) to be h(x). What happens at h(7) and h(-5) (and h(0))?

 

[Corkery]

I'm sorry but I'm really confused. I don't know what to do. I don't know what you want me to do with the "h's" and re-naming the problems

 

[Dick]

Well, what is it YOU want to do? I don't think "find each funstion below (f/g) (x)" is something I know how to do either. What does it mean? Mindscrape seems to be guessing you mean find singularities or denominator factors or asymptotes, but you didn't really ask for that, did you?

 

[Mindscrape]

Alright, let's start off with how a function works. If you have a function f(x), it means that there is some combination of factors of x that has an output (the f(x) part). So, let's look at some arbitrary function

f(x) = x^3 + x^2 + 4x

Depending on the value you put in for x, you will get a different output. The function is dependent on the values of x because the x's tell the function what value it takes on. I could call the function anything.

s(x) = x^3 + x^2 + 4x

a(x) = x^3 + x^2 + 4x

They are all the same f(x)=s(x)=a(x) because the values of x that dictate the outputs are all the same.

If you wanted to multiply the functions, that is perfectly fine. Say

d(x) = x
e(x) = x+1

d(x)*e(x) = x^2 + x

The multiplication produced some new x formula which in turn could serve as its own function.

Let's say that m(x) = x^2 + x
this is equivalent to m(x) = d(x)*e(x)

Back to your the problem
$$\frac{f(x)}{g(x)} = \frac{3}{x(x-7)(x+5)}$$

This is (could be) the answer. What is it you are supposed to do?

What I want to know is completely separate, to see how well you understand functions. f(x)/g(x) makes some new formula with x dependencies, it might as well be h(x). There are some interesting points about this new function h(x), particularly h(7), h(-5), and h(0). What happens there?