Answer :
To determine which pair of functions results in [tex]\((f \circ g)(x) = 12x\)[/tex], we need to compose the functions for each pair and see which one simplifies down to [tex]\(12x\)[/tex]. Let's verify each pair step-by-step.
### Pair 1: [tex]\(f(x) = 3 - 4x\)[/tex] and [tex]\(g(x) = 16x - 3\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = 16x - 3 \][/tex]
[tex]\[ f(g(x)) = f(16x - 3) = 3 - 4(16x - 3) \][/tex]
[tex]\[ = 3 - 64x + 12 \][/tex]
[tex]\[ = 15 - 64x \][/tex]
Clearly, [tex]\((f \circ g)(x) = 15 - 64x\)[/tex], which does not simplify to [tex]\(12x\)[/tex].
### Pair 2: [tex]\(f(x) = 6x^2\)[/tex] and [tex]\(g(x) = \frac{2}{x}\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = \frac{2}{x} \][/tex]
[tex]\[ f(g(x)) = f\left(\frac{2}{x}\right) = 6\left(\frac{2}{x}\right)^2 \][/tex]
[tex]\[ = 6\left(\frac{4}{x^2}\right) \][/tex]
[tex]\[ = \frac{24}{x^2} \][/tex]
Clearly, [tex]\((f \circ g)(x) = \frac{24}{x^2}\)[/tex], which does not simplify to [tex]\(12x\)[/tex].
### Pair 3: [tex]\(f(x) = \sqrt{x}\)[/tex] and [tex]\(g(x) = 144x\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = 144x \][/tex]
[tex]\[ f(g(x)) = f(144x) = \sqrt{144x} \][/tex]
[tex]\[ = 12\sqrt{x} \][/tex]
Clearly, [tex]\((f \circ g)(x) = 12\sqrt{x}\)[/tex], which does not simplify to [tex]\(12x\)[/tex].
### Pair 4: [tex]\(f(x) = 4x\)[/tex] and [tex]\(g(x) = 3x\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = 3x \][/tex]
[tex]\[ f(g(x)) = f(3x) = 4(3x) \][/tex]
[tex]\[ = 12x \][/tex]
Clearly, [tex]\((f \circ g)(x) = 12x\)[/tex], which is exactly what we're looking for.
Therefore, 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]
### Pair 1: [tex]\(f(x) = 3 - 4x\)[/tex] and [tex]\(g(x) = 16x - 3\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = 16x - 3 \][/tex]
[tex]\[ f(g(x)) = f(16x - 3) = 3 - 4(16x - 3) \][/tex]
[tex]\[ = 3 - 64x + 12 \][/tex]
[tex]\[ = 15 - 64x \][/tex]
Clearly, [tex]\((f \circ g)(x) = 15 - 64x\)[/tex], which does not simplify to [tex]\(12x\)[/tex].
### Pair 2: [tex]\(f(x) = 6x^2\)[/tex] and [tex]\(g(x) = \frac{2}{x}\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = \frac{2}{x} \][/tex]
[tex]\[ f(g(x)) = f\left(\frac{2}{x}\right) = 6\left(\frac{2}{x}\right)^2 \][/tex]
[tex]\[ = 6\left(\frac{4}{x^2}\right) \][/tex]
[tex]\[ = \frac{24}{x^2} \][/tex]
Clearly, [tex]\((f \circ g)(x) = \frac{24}{x^2}\)[/tex], which does not simplify to [tex]\(12x\)[/tex].
### Pair 3: [tex]\(f(x) = \sqrt{x}\)[/tex] and [tex]\(g(x) = 144x\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = 144x \][/tex]
[tex]\[ f(g(x)) = f(144x) = \sqrt{144x} \][/tex]
[tex]\[ = 12\sqrt{x} \][/tex]
Clearly, [tex]\((f \circ g)(x) = 12\sqrt{x}\)[/tex], which does not simplify to [tex]\(12x\)[/tex].
### Pair 4: [tex]\(f(x) = 4x\)[/tex] and [tex]\(g(x) = 3x\)[/tex]
First, find [tex]\(f(g(x))\)[/tex]:
[tex]\[ g(x) = 3x \][/tex]
[tex]\[ f(g(x)) = f(3x) = 4(3x) \][/tex]
[tex]\[ = 12x \][/tex]
Clearly, [tex]\((f \circ g)(x) = 12x\)[/tex], which is exactly what we're looking for.
Therefore, 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]
