To determine for which pair of functions [tex]\((f \circ g)(x) = 12x\)[/tex], we need to compose the given pairs of functions and verify if the resulting function equals [tex]\(12x\)[/tex]. Let's go through each pair step-by-step.
### Pair 1: [tex]\( f(x) = 3 - 4x \)[/tex] and [tex]\( g(x) = 16x - 3 \)[/tex]
First, we find [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f(16x - 3) \][/tex]
[tex]\[ f(16x - 3) = 3 - 4(16x - 3) \][/tex]
[tex]\[ = 3 - 64x + 12 \][/tex]
[tex]\[ = 15 - 64x \][/tex]
This is not equal to [tex]\(12x\)[/tex].
### Pair 2: [tex]\( f(x) = 6x^2 \)[/tex] and [tex]\( g(x) = \frac{2}{x} \)[/tex]
Next, we find [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{2}{x}\right) \][/tex]
[tex]\[ f\left(\frac{2}{x}\right) = 6\left(\frac{2}{x}\right)^2 \][/tex]
[tex]\[ = 6 \cdot \left(\frac{4}{x^2}\right) \][/tex]
[tex]\[ = \frac{24}{x^2} \][/tex]
This is not equal to [tex]\(12x\)[/tex].
### Pair 3: [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) = 144x \)[/tex]
Now, we find [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f(144x) \][/tex]
[tex]\[ f(144x) = \sqrt{144x} \][/tex]
[tex]\[ = 12\sqrt{x} \][/tex]
This is not equal to [tex]\(12x\)[/tex].
### Pair 4: [tex]\( f(x) = 4x \)[/tex] and [tex]\( g(x) = 3x \)[/tex]
Finally, we find [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(g(x)) = f(3x) \][/tex]
[tex]\[ f(3x) = 4(3x) \][/tex]
[tex]\[ = 12x \][/tex]
This is indeed equal to [tex]\(12x\)[/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]
