Alright, let's solve this step by step for each part:
### Part (a): Finding [tex]\((f \circ g)(x)\)[/tex]
To find [tex]\((f \circ g)(x)\)[/tex], we need to compose the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex], which means applying [tex]\(g(x)\)[/tex] first and then applying [tex]\(f(x)\)[/tex] to the result.
Given:
[tex]\[ f(x) = 5x - 2 \][/tex]
[tex]\[ g(x) = 3x^2 - 4 \][/tex]
The composition [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[ (f \circ g)(x) = f(g(x)) \][/tex]
First, calculate [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = 3x^2 - 4 \][/tex]
Then, substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f(3x^2 - 4) \][/tex]
Now, apply the function [tex]\(f\)[/tex] to [tex]\((3x^2 - 4)\)[/tex]:
[tex]\[ f(3x^2 - 4) = 5(3x^2 - 4) - 2 \][/tex]
[tex]\[ = 15x^2 - 20 - 2 \][/tex]
[tex]\[ = 15x^2 - 22 \][/tex]
Thus, [tex]\((f \circ g)(x) = 15x^2 - 22\)[/tex].
### Part (b): Finding [tex]\((g \circ f)(x)\)[/tex]
To find [tex]\((g \circ f)(x)\)[/tex], we need to compose the functions [tex]\(g\)[/tex] and [tex]\(f\)[/tex], which means applying [tex]\(f(x)\)[/tex] first and then applying [tex]\(g(x)\)[/tex] to the result.
Given:
[tex]\[ f(x) = 5x - 2 \][/tex]
[tex]\[ g(x) = 3x^2 - 4 \][/tex]
The composition [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[ (g \circ f)(x) = g(f(x)) \][/tex]
First, calculate [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 5x - 2 \][/tex]
Then, substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[ g(f(x)) = g(5x - 2) \][/tex]
Now, apply the function [tex]\(g\)[/tex] to [tex]\((5x - 2)\)[/tex]:
[tex]\[ g(5x - 2) = 3(5x - 2)^2 - 4 \][/tex]
Expand [tex]\((5x - 2)^2\)[/tex]:
[tex]\[ (5x - 2)^2 = (5x - 2)(5x - 2) \][/tex]
[tex]\[ = 25x^2 - 20x - 20x + 4 \][/tex]
[tex]\[ = 25x^2 - 20x + 4 \][/tex]
Now apply [tex]\(g\)[/tex] to the expanded expression:
[tex]\[ g(5x - 2) = 3(25x^2 - 20x + 4) - 4 \][/tex]
[tex]\[ = 75x^2 - 60x + 12 - 4 \][/tex]
[tex]\[ = 75x^2 - 60x + 8 \][/tex]
Thus, [tex]\((g \circ f)(x) = 75x^2 - 60x + 8\)[/tex].
### Part (c): Finding [tex]\((f \circ g)(2)\)[/tex]
To find [tex]\((f \circ g)(2)\)[/tex], we substitute [tex]\(x = 2\)[/tex] into our expression for [tex]\((f \circ g)(x)\)[/tex].
From part (a):
[tex]\[ (f \circ g)(x) = 15x^2 - 22 \][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ (f \circ g)(2) = 15(2)^2 - 22 \][/tex]
[tex]\[ = 15(4) - 22 \][/tex]
[tex]\[ = 60 - 22 \][/tex]
[tex]\[ = 38 \][/tex]
Thus, [tex]\((f \circ g)(2) = 38\)[/tex].
### Part (d): Finding [tex]\((g \circ f)(2)\)[/tex]
To find [tex]\((g \circ f)(2)\)[/tex], we substitute [tex]\(x = 2\)[/tex] into our expression for [tex]\((g \circ f)(x)\)[/tex].
From part (b):
[tex]\[ (g \circ f)(x) = 75x^2 - 60x + 8 \][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[ (g \circ f)(2) = 75(2)^2 - 60(2) + 8 \][/tex]
[tex]\[ = 75(4) - 120 + 8 \][/tex]
[tex]\[ = 300 - 120 + 8 \][/tex]
[tex]\[ = 180 + 8 \][/tex]
[tex]\[ = 188 \][/tex]
Thus, [tex]\((g \circ f)(2) = 188\)[/tex].
### Summary:
a. [tex]\((f \circ g)(x) = 15x^2 - 22\)[/tex]
b. [tex]\((g \circ f)(x) = 75x^2 - 60x + 8\)[/tex]
c. [tex]\((f \circ g)(2) = 38\)[/tex]
d. [tex]\((g \circ f)(2) = 188\)[/tex]
