How Do You Solve Composite Functions and Their Domains?

[dragon5]

Homework Statement

need some help with my h/w, stuck on these problems, please explain the process and how you obtained the answer, thank you!


1. Let f(x)=x^3 - 4 and g(x)= 3rd root of x + 7

-find the composite function (fog)(x) and determine its domaine and range

-Find the composite function (gof)(x) and determine its domain and range


2. Given the function f(x)= 2x+1/x-3

-find the domaine and range of f(x)
-Find the inverse function f^-1(x) and determine its domain and range
-Graph f(x) and f^-1(x)


3. Given the rational function f(x)= x^2+12/x-2
-graph y=f(x)
-determine the domain and range of f(x)
-decide whether the function is even, odd, or neither
-find the horizontal and vertical intercepts of f(x)
-list the asymptotes of f(x)

Homework Equations

The Attempt at a Solution

First problem I am puzzled by on a few things. I know f o g means plug f(x) into g(x) and solve that way, so 3rd root of (x^3-4+7)=3rd root of (x^3-11) correct? I am confused as to where to go from there.

Then vice versa on that problem with g o f, 3rd root of (x+7) to the 3rd power, the 3's cancel so (x+7-4) so x = -3??


I know domain and range are values on the coordinate plane from which the graph extends to, domain is the x value at which the graph is limited, and range the y value.


The problem is I am unsure of how to find the domain/range, or the correct formula to use to find it.


2. On number 2 I graphed using my calculator and pretty much guessed the domain being (-(infinity), 3) U (3, infinity) Range being (-infinity, -1/3)

 

[Mark44]

Homework Statement

need some help with my h/w, stuck on these problems, please explain the process and how you obtained the answer, thank you!

You're new to this forum, so you might not have read the rules, which are posted here: https://www.physicsforums.com/showthread.php?t=5374
One section of the rules governs the kind of help you can expect.
Homework Help (excerpt)
On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given. Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made.

1. Let f(x)=x^3 - 4 and g(x)= 3rd root of x + 7

-find the composite function (fog)(x) and determine its domaine and range

-Find the composite function (gof)(x) and determine its domain and range


2. Given the function f(x)= 2x+1/x-3

-find the domaine and range of f(x)
-Find the inverse function f^-1(x) and determine its domain and range
-Graph f(x) and f^-1(x)


3. Given the rational function f(x)= x^2+12/x-2
-graph y=f(x)
-determine the domain and range of f(x)
-decide whether the function is even, odd, or neither
-find the horizontal and vertical intercepts of f(x)
-list the asymptotes of f(x)

Homework Equations

The Attempt at a Solution

First problem I am puzzled by on a few things. I know f o g means plug f(x) into g(x) and solve that way, so 3rd root of (x^3-4+7)=3rd root of (x^3-11) correct? I am confused as to where to go from there.

No, f o g doesn't mean plug f(x) into g(x). (f o g)(x) = f(g(x)), so it's the other way around from what you said.

x^3 - 4 + 7 \neq x^3 - 11. Check your arithmetic.

Then vice versa on that problem with g o f, 3rd root of (x+7) to the 3rd power, the 3's cancel so (x+7-4) so x = -3??

The goal is not to solve an equation here, but to find a formula for (g o f)(x) = g(f(x)).

I know domain and range are values on the coordinate plane from which the graph extends to, domain is the x value at which the graph is limited, and range the y value.


The problem is I am unsure of how to find the domain/range, or the correct formula to use to find it.


2. On number 2 I graphed using my calculator and pretty much guessed the domain being (-(infinity), 3) U (3, infinity) Range being (-infinity, -1/3)

You have the domain right, but not the range.