Certainly! Here, we are given two functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. To find [tex]\((g \circ f)(x)\)[/tex], we need to evaluate [tex]\( g(f(x)) \)[/tex].
First, let's start with the given functions:
[tex]\[ f(x) = 6x - 4 \][/tex]
[tex]\[ g(x) = 3x^2 - 2x - 10 \][/tex]
To find [tex]\( (g \circ f)(x) \)[/tex], we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]. That means we replace every [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex].
Starting with [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = 6x - 4 \][/tex]
Next, substitute [tex]\( 6x - 4 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[ g(6x - 4) \][/tex]
Now, replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[ g(6x - 4) = 3(6x - 4)^2 - 2(6x - 4) - 10 \][/tex]
Next, let's expand and simplify this expression step by step.
First, expand [tex]\( (6x - 4)^2 \)[/tex]:
[tex]\[ (6x - 4)^2 = (6x - 4)(6x - 4) = 36x^2 - 48x + 16 \][/tex]
Now, multiply by 3:
[tex]\[ 3(36x^2 - 48x + 16) = 108x^2 - 144x + 48 \][/tex]
Next, distribute -2 through [tex]\( 6x - 4 \)[/tex]:
[tex]\[ -2(6x - 4) = -12x + 8 \][/tex]
Now, combine all parts:
[tex]\[ 108x^2 - 144x + 48 - 12x + 8 - 10 \][/tex]
Finally, combine like terms:
[tex]\[ 108x^2 - 144x - 12x + 48 + 8 - 10 \][/tex]
[tex]\[ 108x^2 - 156x + 46 \][/tex]
Therefore, [tex]\((g \circ f)(x) = 108x^2 - 156x + 46\)[/tex].
