Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

Algebra - MATH 1115 SU24

Given [tex]\( f(x) = 4x^3 \)[/tex] and [tex]\( g(x) = x^2 + 4x - 3 \)[/tex], find each of the following. Identify the domain of each.

(a) [tex]\( (f \circ g)(x) \)[/tex]

(b) [tex]\( (g \circ f)(x) \)[/tex]

(c) [tex]\( (f \circ f)(x) \)[/tex]

(a) [tex]\( (f \circ g)(x) = \square \)[/tex] (Simplify your answer.)


Sagot :

Let's solve the problem step-by-step.

### Given Functions:
[tex]\[ f(x) = 4x^3 \][/tex]
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]

### Part (a): [tex]\((f \circ g)(x)\)[/tex]
The notation [tex]\((f \circ g)(x)\)[/tex] means we want to find [tex]\(f(g(x))\)[/tex].

1. First, we express [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]

2. Next, we substitute [tex]\(g(x)\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f(x^2 + 4x - 3) \][/tex]

3. Now, use the definition of [tex]\(f\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
So we need to find [tex]\(f(x^2 + 4x - 3)\)[/tex].

4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(f(x) = 4x^3\)[/tex] with [tex]\(x^2 + 4x - 3\)[/tex]:
[tex]\[ f(x^2 + 4x - 3) = 4(x^2 + 4x - 3)^3 \][/tex]

So the solution for [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[ (f \circ g)(x) = 4(x^2 + 4x - 3)^3 \][/tex]

### Part (b): [tex]\((g \circ f)(x)\)[/tex]
The notation [tex]\((g \circ f)(x)\)[/tex] means we want to find [tex]\(g(f(x))\)[/tex].

1. First, we express [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]

2. Next, we substitute [tex]\(f(x)\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[ g(f(x)) = g(4x^3) \][/tex]

3. Now, use the definition of [tex]\(g\)[/tex]:
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]
So we need to find [tex]\(g(4x^3)\)[/tex].

4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(g(x) = x^2 + 4x - 3\)[/tex] with [tex]\(4x^3\)[/tex]:
[tex]\[ g(4x^3) = (4x^3)^2 + 4(4x^3) - 3 \][/tex]
[tex]\[ g(4x^3) = 16x^6 + 16x^3 - 3 \][/tex]

So the solution for [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[ (g \circ f)(x) = 16x^6 + 16x^3 - 3 \][/tex]

### Part (c): [tex]\((f \circ f)(x)\)[/tex]
The notation [tex]\((f \circ f)(x)\)[/tex] means we want to find [tex]\(f(f(x))\)[/tex].

1. First, we express [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]

2. Next, we substitute [tex]\(f(x)\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(f(x)) = f(4x^3) \][/tex]

3. Now, use the definition of [tex]\(f\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
So we need to find [tex]\(f(4x^3)\)[/tex].

4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(f(x) = 4x^3\)[/tex] with [tex]\(4x^3\)[/tex]:
[tex]\[ f(4x^3) = 4(4x^3)^3 \][/tex]
[tex]\[ f(4x^3) = 4(64x^9) = 256x^9 \][/tex]

So the solution for [tex]\((f \circ f)(x)\)[/tex] is:
[tex]\[ (f \circ f)(x) = 256x^9 \][/tex]

### Domain of Each Composite Function:
1. [tex]\( (f \circ g)(x) \)[/tex]:
The domain of [tex]\(g(x)\)[/tex] is all real numbers because it is a polynomial. The output of [tex]\(g(x)\)[/tex] must also be within the domain of [tex]\(f(x)\)[/tex], which is all real numbers for [tex]\(4x^3\)[/tex]. Thus, the domain of [tex]\((f \circ g)(x)\)[/tex] is all real numbers.

2. [tex]\( (g \circ f)(x) \)[/tex]:
The domain of [tex]\(f(x)\)[/tex] is all real numbers. The output of [tex]\(f(x)\)[/tex] must be within the domain of [tex]\(g(x)\)[/tex], which is all real numbers for [tex]\(x^2 + 4x - 3\)[/tex]. Thus, the domain of [tex]\((g \circ f)(x)\)[/tex] is all real numbers.

3. [tex]\( (f \circ f)(x) \)[/tex]:
The domain of [tex]\(f(x)\)[/tex] is all real numbers. The output of [tex]\(f(x)\)[/tex] must be within the domain of itself, which is all real numbers for [tex]\(4x^3\)[/tex]. Thus, the domain of [tex]\((f \circ f)(x)\)[/tex] is all real numbers.

So, the domains for [tex]\((f \circ g)(x)\)[/tex], [tex]\((g \circ f)(x)\)[/tex], and [tex]\((f \circ f)(x)\)[/tex] are all real numbers, which can be written as:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]