Help with composite functions?

[Nekou19]

Hello,
I have a calculus problem involving composite functions, and I am not too sure where to begin. Here is the question:

Q: You have four functions: f(v) = (v^2 - 3v)^8
g(u) = 2(sqrt)(u + 1)
h(x) = 1 / x+1
w= f o g o h (That is, w(x) is the composite function (f o g o h)(x) ) (the "o" represents "circle")

1) What is the domain of w?

I think i have to first find g(h(x)), by plugging in h(x) into the g(u) equation. Then i should take what i get for g(h(x)), and plug it into f(v)? But I am not sure if this is the correct method. If it the correct method to start, I would really appreciate any advise on the next steps.

Thank You!

 

[Idealistic]

The good thing about domains and composite functions is that you don't even really need to determine what w(x) looks explicitly, just look at where it's component functions are continuos.

f is continuos on R,
g is continuos on [-1, infinity),
h is continuos on (-infinity, -1) U (-1, infinity)

so,

w is continuos on (-1, infinity)

 

[lookagain]

The good thing about domains and composite functions is that you don't even
really need to determine what w(x) looks explicitly, just look at where it's component functions
are continuous.

f is continuos on R,
g is continuos on [-1, infinity),
h is continuos on (-infinity, -1) U (-1, infinity)

so,

\(\displaystyle > \ > \ >\)w is continuos on (-1, infinity)\(\displaystyle < \ < \ <\)

Edit: Link removed

For h(x), x is not equal to -1.

We don't look set the (u + 1) in g(u + 1) equal to zero.

We look at g(1/(x + 1) + 1) = g((x + 2)/(x + 1))

This leads to x <= -2 or x > -1.

And f(g((x + 2)(x + 1))) has no further restrictions on the x-values.


Then the domain is \(\displaystyle (-\infty, -2] \ U \ (-1, \infty)\)

Idealistic,

for example, check x = -3:

h(-3) = -1/2

\(\displaystyle Edit:\)

g(h(-3)) = g(-1/2) = 2sqrt(-1/2 + 1) = 2sqrt(1/2)

w = f(g(h(-3))) = f(2sqrt(1/2)), which has no further restrictions on x.

So x = -3 is in the domain.


Idealistic,

you left out the portion of the domain, x <= -2.

 

[Idealistic]

I was under the impression that g was irrational,

g = 2*(u + 1)^(1/2)

so u could not be less that -1.

EDIT

How foolish, I read the post and was thinking h(g(x)) for some reason.