Alright, let's dive into the question step by step.
Given the functions:
[tex]\[ f(x) = 2x - 3 \][/tex]
[tex]\[ g(x) = 3x^2 - 4 \][/tex]
We need to find:
a. The composition [tex]\((f \circ g)(x)\)[/tex]
b. The composition [tex]\((g \circ f)(x)\)[/tex]
c. The value of [tex]\((f \circ g)(2)\)[/tex]
### a. Finding [tex]\((f \circ g)(x)\)[/tex]
The composition [tex]\((f \circ g)(x)\)[/tex] means we want to apply [tex]\(g(x)\)[/tex] first and then apply [tex]\(f\)[/tex] to the result of [tex]\(g(x)\)[/tex].
Start by calculating [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = 3x^2 - 4 \][/tex]
Now, apply [tex]\(f\)[/tex] to [tex]\(g(x)\)[/tex]:
[tex]\[ (f \circ g)(x) = f(g(x)) = f(3x^2 - 4) \][/tex]
Substitute [tex]\(3x^2 - 4\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(3x^2 - 4) = 2(3x^2 - 4) - 3 \][/tex]
[tex]\[ = 6x^2 - 8 - 3 \][/tex]
[tex]\[ = 6x^2 - 11 \][/tex]
So, [tex]\((f \circ g)(x) = 6x^2 - 11\)[/tex].
### b. Finding [tex]\((g \circ f)(x)\)[/tex]
The composition [tex]\((g \circ f)(x)\)[/tex] means we want to apply [tex]\(f(x)\)[/tex] first and then apply [tex]\(g\)[/tex] to the result of [tex]\(f(x)\)[/tex].
Start by calculating [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 2x - 3 \][/tex]
Now, apply [tex]\(g\)[/tex] to [tex]\(f(x)\)[/tex]:
[tex]\[ (g \circ f)(x) = g(f(x)) = g(2x - 3) \][/tex]
Substitute [tex]\(2x - 3\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[ g(2x - 3) = 3(2x - 3)^2 - 4 \][/tex]
First, expand [tex]\((2x - 3)^2\)[/tex]:
[tex]\[ (2x - 3)^2 = 4x^2 - 12x + 9 \][/tex]
Then, substitute back into [tex]\(g(x)\)[/tex]:
[tex]\[ g(2x - 3) = 3(4x^2 - 12x + 9) - 4 \][/tex]
[tex]\[ = 12x^2 - 36x + 27 - 4 \][/tex]
[tex]\[ = 12x^2 - 36x + 23 \][/tex]
So, [tex]\((g \circ f)(x) = 12x^2 - 36x + 23\)[/tex].
### c. Finding [tex]\((f \circ g)(2)\)[/tex]
Now, we need to find the value of [tex]\((f \circ g)(2)\)[/tex]:
Start by calculating [tex]\(g(2)\)[/tex]:
[tex]\[ g(2) = 3(2)^2 - 4 \][/tex]
[tex]\[ = 3(4) - 4 \][/tex]
[tex]\[ = 12 - 4 \][/tex]
[tex]\[ = 8 \][/tex]
Next, use this result in [tex]\(f\)[/tex]:
[tex]\[ f(g(2)) = f(8) \][/tex]
[tex]\[ f(8) = 2(8) - 3 \][/tex]
[tex]\[ = 16 - 3 \][/tex]
[tex]\[ = 13 \][/tex]
So, [tex]\((f \circ g)(2) = 13\)[/tex].
### Summary:
a. [tex]\((f \circ g)(x) = 6x^2 - 11\)[/tex]
b. [tex]\((g \circ f)(x) = 12x^2 - 36x + 23\)[/tex]
c. [tex]\((f \circ g)(2) = 13\)[/tex]
