Question

Consider the following. f(x) = x3 + 4, g(x) = 3x (a) Find the function (f ∘ g)(x). (f ∘ g)(x) = Find the domain of (f ∘ g)(x). (Enter your answer using interval notation.) (b) Find the function (g ∘ f)(x). (g ∘ f)(x) = Find the domain of (g ∘ f)(x). (Enter your answer using interval notation.) (c) Find the function (f ∘ f)(x). (f ∘ f)(x) = Find the domain of (f ∘ f)(x). (Enter your answer using interval notation.) (d) Find the function (g ∘ g)(x). (g ∘ g)(x) = Find the domain of (g ∘ g)(x). (Enter your answer using interval notation.)

          Consider the following.
f(x) = x3 + 4, g(x) = 3x
(a) Find the function (f ∘ g)(x).
(f ∘ g)(x) = 
Find the domain of (f ∘ g)(x). (Enter your answer using interval notation.)
(b) Find the function (g ∘ f)(x).
(g ∘ f)(x) = 
Find the domain of (g ∘ f)(x). (Enter your answer using interval notation.)
(c) Find the function (f ∘ f)(x).
(f ∘ f)(x) = 
Find the domain of (f ∘ f)(x). (Enter your answer using interval notation.)
(d) Find the function (g ∘ g)(x).
(g ∘ g)(x) = 
Find the domain of (g ∘ g)(x). (Enter your answer using interval notation.)

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Added by Walter G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Bailey Cooper

09/03/2022

Step 1

Step 1:** Given functions: \(f(x) = x^3 + 4\) \(g(x) = 3x\) (a) Find \((f \circ g)(x)\): \((f \circ g)(x) = f(g(x)) = f(3x) = (3x)^3 + 4 = 27x^3 + 4\) Domain of \((f \circ g)(x)\): \([- \infty, + \infty]\) (b) Find \((g \circ f)(x)\): Show more…

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Consider the following. f(x) = x3 + 4, g(x) = 3x (a) Find the function (f ∘ g)(x). (f ∘ g)(x) = Find the domain of (f ∘ g)(x). (Enter your answer using interval notation.) (b) Find the function (g ∘ f)(x). (g ∘ f)(x) = Find the domain of (g ∘ f)(x). (Enter your answer using interval notation.) (c) Find the function (f ∘ f)(x). (f ∘ f)(x) = Find the domain of (f ∘ f)(x). (Enter your answer using interval notation.) (d) Find the function (g ∘ g)(x). (g ∘ g)(x) = Find the domain of (g ∘ g)(x). (Enter your answer using interval notation.)

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00:01 Okay, so for this question, we are given two functions.

00:04 We are given f of x equals x cubed plus four, and we're given g of x equals 3x.

00:29 So first thing we're asked to find is f of g of x, which basically means that we're going to substitute g of x or 3x for x in, f of x.

00:46 So what that's going to look like is 3x cubed plus four.

00:56 That's all it is.

00:59 And then it asks us to find the domain of f of g of x, which basically just means what are all of the reasonable values that we could substitute for x? are there any values that are going to make this undefined or cause any other issues for us.

01:21 And i would say, no, i mean, we can cube any number, right? we can cube negative 1.

01:28 We can cube negative 10 million.

01:30 We can cube 0.

01:31 We can cube positive 10 million and multiply it by three and add four to it.

01:36 So i think the domain for this is just going to be negative infinity to infinity.

01:42 You can pretty much cube anything.

01:46 Okay.

01:49 Next, they don't want us to find range, which is interesting.

01:54 But that's fine.

01:54 We can move on.

01:56 Next, they want us to find the function g of f of x.

02:01 So let's do that...

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