Answer :
To determine for which pair of functions [tex]\((f \circ g)(x) = 12x\)[/tex], we need to check each pair by composing [tex]\(f(x)\)[/tex] with [tex]\(g(x)\)[/tex] and see if the result equals [tex]\(12x\)[/tex].
Let's go through each pair one by one:
1. Pair: [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]
Substitute [tex]\(16x - 3\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(16x - 3) = 3 - 4(16x - 3) \][/tex]
[tex]\[ f(16x - 3) = 3 - 64x + 12 \][/tex]
[tex]\[ f(16x - 3) = 15 - 64x \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ 15 - 64x \neq 12x \][/tex]
So, this pair does not satisfy the equation.
2. Pair: [tex]\(f(x) = 6x^2\)[/tex] and [tex]\(g(x) = \frac{2}{x}\)[/tex]
Now, we find [tex]\(f(g(x))\)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{2}{x}\right) \][/tex]
Substitute [tex]\(\frac{2}{x}\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f\left(\frac{2}{x}\right) = 6\left(\frac{2}{x}\right)^2 \][/tex]
[tex]\[ f\left(\frac{2}{x}\right) = 6 \cdot \frac{4}{x^2} \][/tex]
[tex]\[ f\left(\frac{2}{x}\right) = \frac{24}{x^2} \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ \frac{24}{x^2} \neq 12x \][/tex]
So, this pair does not satisfy the equation.
3. Pair: [tex]\(f(x) = \sqrt{x}\)[/tex] and [tex]\(g(x) = 144x\)[/tex]
Next, we find [tex]\(f(g(x))\)[/tex]:
[tex]\[ f(g(x)) = f(144x) \][/tex]
Substitute [tex]\(144x\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(144x) = \sqrt{144x} \][/tex]
[tex]\[ f(144x) = 12\sqrt{x} \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ 12\sqrt{x} \neq 12x \][/tex]
So, this pair does not satisfy the equation.
4. Pair: [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]
Substitute [tex]\(3x\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(3x) = 4(3x) \][/tex]
[tex]\[ f(3x) = 12x \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ 12x = 12x \][/tex]
This pair satisfies the equation.
Conclusion:
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].
Let's go through each pair one by one:
1. Pair: [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]
Substitute [tex]\(16x - 3\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(16x - 3) = 3 - 4(16x - 3) \][/tex]
[tex]\[ f(16x - 3) = 3 - 64x + 12 \][/tex]
[tex]\[ f(16x - 3) = 15 - 64x \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ 15 - 64x \neq 12x \][/tex]
So, this pair does not satisfy the equation.
2. Pair: [tex]\(f(x) = 6x^2\)[/tex] and [tex]\(g(x) = \frac{2}{x}\)[/tex]
Now, we find [tex]\(f(g(x))\)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{2}{x}\right) \][/tex]
Substitute [tex]\(\frac{2}{x}\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f\left(\frac{2}{x}\right) = 6\left(\frac{2}{x}\right)^2 \][/tex]
[tex]\[ f\left(\frac{2}{x}\right) = 6 \cdot \frac{4}{x^2} \][/tex]
[tex]\[ f\left(\frac{2}{x}\right) = \frac{24}{x^2} \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ \frac{24}{x^2} \neq 12x \][/tex]
So, this pair does not satisfy the equation.
3. Pair: [tex]\(f(x) = \sqrt{x}\)[/tex] and [tex]\(g(x) = 144x\)[/tex]
Next, we find [tex]\(f(g(x))\)[/tex]:
[tex]\[ f(g(x)) = f(144x) \][/tex]
Substitute [tex]\(144x\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(144x) = \sqrt{144x} \][/tex]
[tex]\[ f(144x) = 12\sqrt{x} \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ 12\sqrt{x} \neq 12x \][/tex]
So, this pair does not satisfy the equation.
4. Pair: [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]
Substitute [tex]\(3x\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(3x) = 4(3x) \][/tex]
[tex]\[ f(3x) = 12x \][/tex]
We compare this with [tex]\(12x\)[/tex]:
[tex]\[ 12x = 12x \][/tex]
This pair satisfies the equation.
Conclusion:
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].
