Let's analyze and calculate each part step-by-step using the functions [tex]\( f(x) = x - 3 \)[/tex] and [tex]\( g(x) = 5x^2 - 2 \)[/tex].
### Part (a): [tex]\((f \circ g)(x)\)[/tex]
The composition of two functions [tex]\( (f \circ g)(x) \)[/tex] means we first apply [tex]\( g(x) \)[/tex] and then apply [tex]\( f \)[/tex] to the result. In other words, [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex].
Given:
[tex]\[ g(x) = 5x^2 - 2 \][/tex]
Now, apply [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]:
[tex]\[ f(g(x)) = f(5x^2 - 2) \][/tex]
Since [tex]\( f(x) = x - 3 \)[/tex], replace [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( 5x^2 - 2 \)[/tex]:
[tex]\[ f(5x^2 - 2) = (5x^2 - 2) - 3 \][/tex]
[tex]\[ f(5x^2 - 2) = 5x^2 - 5 \][/tex]
So:
[tex]\[ (f \circ g)(x) = 5x^2 - 5 \][/tex]
### Part (b): [tex]\((g \circ f)(x)\)[/tex]
The composition of two functions [tex]\( (g \circ f)(x) \)[/tex] means we first apply [tex]\( f(x) \)[/tex] and then apply [tex]\( g \)[/tex] to the result. In other words, [tex]\( (g \circ f)(x) = g(f(x)) \)[/tex].
Given:
[tex]\[ f(x) = x - 3 \][/tex]
Now, apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x - 3) \][/tex]
Since [tex]\( g(x) = 5x^2 - 2 \)[/tex], replace [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( x - 3 \)[/tex]:
[tex]\[ g(x - 3) = 5(x - 3)^2 - 2 \][/tex]
[tex]\[ g(x - 3) = 5(x^2 - 6x + 9) - 2 \][/tex]
[tex]\[ g(x - 3) = 5x^2 - 30x + 45 - 2 \][/tex]
[tex]\[ g(x - 3) = 5x^2 - 30x + 43 \][/tex]
So:
[tex]\[ (g \circ f)(x) = 5x^2 - 30x + 43 \][/tex]
### Part (c): [tex]\((f \circ g)(1)\)[/tex]
To find [tex]\((f \circ g)(1)\)[/tex], we first find [tex]\( g(1) \)[/tex] and then apply [tex]\( f \)[/tex] to the result.
First, calculate [tex]\( g(1) \)[/tex]:
[tex]\[ g(1) = 5 \cdot 1^2 - 2 \][/tex]
[tex]\[ g(1) = 5 - 2 \][/tex]
[tex]\[ g(1) = 3 \][/tex]
Now, apply [tex]\( f \)[/tex] to the result:
[tex]\[ f(g(1)) = f(3) \][/tex]
[tex]\[ f(3) = 3 - 3 \][/tex]
[tex]\[ f(3) = 0 \][/tex]
So:
[tex]\[ (f \circ g)(1) = 0 \][/tex]
### Part (d): [tex]\((g \circ f)(1)\)[/tex]
To find [tex]\((g \circ f)(1)\)[/tex], we first find [tex]\( f(1) \)[/tex] and then apply [tex]\( g \)[/tex] to the result.
First, calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 1 - 3 \][/tex]
[tex]\[ f(1) = -2 \][/tex]
Now, apply [tex]\( g \)[/tex] to the result:
[tex]\[ g(f(1)) = g(-2) \][/tex]
[tex]\[ g(-2) = 5 \cdot (-2)^2 - 2 \][/tex]
[tex]\[ g(-2) = 5 \cdot 4 - 2 \][/tex]
[tex]\[ g(-2) = 20 - 2 \][/tex]
[tex]\[ g(-2) = 18 \][/tex]
So:
[tex]\[ (g \circ f)(1) = 18 \][/tex]
### Summary
The functions and their evaluations are as follows:
a. [tex]\((f \circ g)(x) = 5x^2 - 5\)[/tex]
b. [tex]\((g \circ f)(x) = 5x^2 - 30x + 43\)[/tex]
c. [tex]\((f \circ g)(1) = 0\)[/tex]
d. [tex]\((g \circ f)(1) = 18\)[/tex]
