To find which pair of functions results in [tex]\((f \circ g)(x) = 12x\)[/tex], we must evaluate the composition [tex]\( (f \circ g)(x) \)[/tex] for each pair.
### 1. Pair [tex]\( f(x) = 3 - 4x \)[/tex] and [tex]\( g(x) = 16x - 3 \)[/tex]
To find [tex]\( (f \circ g)(x) \)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(16x - 3)
\][/tex]
Now, substitute [tex]\( 16x - 3 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(16x - 3) = 3 - 4(16x - 3) = 3 - 64x + 12 = 15 - 64x
\][/tex]
Clearly, [tex]\( 15 - 64x \neq 12x \)[/tex].
### 2. Pair [tex]\( f(x) = 6x^2 \)[/tex] and [tex]\( g(x) = \frac{2}{x} \)[/tex]
To find [tex]\( (f \circ g)(x) \)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f\left( \frac{2}{x} \right)
\][/tex]
Now, substitute [tex]\( \frac{2}{x} \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f\left( \frac{2}{x} \right) = 6 \left( \frac{2}{x} \right)^2 = 6 \left( \frac{4}{x^2} \right) = \frac{24}{x^2}
\][/tex]
Clearly, [tex]\( \frac{24}{x^2} \neq 12x \)[/tex].
### 3. Pair [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) = 144x \)[/tex]
To find [tex]\( (f \circ g)(x) \)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(144x)
\][/tex]
Now, substitute [tex]\( 144x \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(144x) = \sqrt{144x} = 12\sqrt{x}
\][/tex]
Clearly, [tex]\( 12\sqrt{x} \neq 12x \)[/tex].
### 4. Pair [tex]\( f(x) = 4x \)[/tex] and [tex]\( g(x) = 3x \)[/tex]
To find [tex]\( (f \circ g)(x) \)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(3x)
\][/tex]
Now, substitute [tex]\( 3x \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(3x) = 4(3x) = 12x
\][/tex]
Clearly, [tex]\( 12x = 12x \)[/tex].
Thus, the pair of functions for which [tex]\( (f \circ g)(x) = 12x \)[/tex] is:
[tex]\[
\boxed{f(x) = 4x \text{ and } g(x) = 3x}
\][/tex]
