To find out which pair of functions gives us [tex]\((f \circ g)(x) = 12x\)[/tex], we need to compose each pair of functions and check if the resulting composition is equal to [tex]\(12x\)[/tex].
Let's go through each given pair of functions step by step:
1. First Pair:
[tex]\[
f(x) = 3 - 4x \quad \text{and} \quad g(x) = 16x - 3
\][/tex]
We need to compute [tex]\((f \circ g)(x)\)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(16x - 3)
\][/tex]
Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(16x - 3) = 3 - 4(16x - 3) = 3 - 64x + 12 = 15 - 64x
\][/tex]
Check if [tex]\(15 - 64x = 12x\)[/tex]:
[tex]\[
15 - 64x \neq 12x
\][/tex]
This pair does not satisfy [tex]\((f \circ g)(x) = 12x\)[/tex].
2. Second Pair:
[tex]\[
f(x) = 6x^2 \quad \text{and} \quad g(x) = \frac{2}{x}
\][/tex]
We need to compute [tex]\((f \circ g)(x)\)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f\left(\frac{2}{x}\right)
\][/tex]
Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f\left(\frac{2}{x}\right) = 6 \left(\frac{2}{x}\right)^2 = 6 \cdot \frac{4}{x^2} = \frac{24}{x^2}
\][/tex]
Check if [tex]\(\frac{24}{x^2} = 12x\)[/tex]:
[tex]\[
\frac{24}{x^2} \neq 12x
\][/tex]
This pair does not satisfy [tex]\((f \circ g)(x) = 12x\)[/tex].
3. Third Pair:
[tex]\[
f(x) = \sqrt{x} \quad \text{and} \quad g(x) = 144x
\][/tex]
We need to compute [tex]\((f \circ g)(x)\)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(144x)
\][/tex]
Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(144x) = \sqrt{144x} = 12\sqrt{x}
\][/tex]
Check if [tex]\(12\sqrt{x} = 12x\)[/tex]:
[tex]\[
12\sqrt{x} \quad \text{is equal to} \quad 12x \quad \text{when} \quad x=1
\][/tex]
This pair satisfies [tex]\((f \circ g)(x) = 12x\)[/tex] when [tex]\(x=1\)[/tex].
4. Fourth Pair:
[tex]\[
f(x) = 4x \quad \text{and} \quad g(x) = 3x
\][/tex]
We need to compute [tex]\((f \circ g)(x)\)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(3x)
\][/tex]
Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[
f(3x) = 4(3x) = 12x
\][/tex]
Check if [tex]\(12x = 12x\)[/tex]:
[tex]\[
12x = 12x
\][/tex]
This pair satisfies [tex]\((f \circ g)(x) = 12x\)[/tex].
Therefore, the pair of functions for which [tex]\((f \circ g)(x) = 12x\)[/tex] is:
- [tex]\( f(x) = 4x \)[/tex] and [tex]\( g(x) = 3x \)[/tex]
- [tex]\( f(x) = \sqrt{x} \)[/tex] and [tex]\( g(x) = 144x \)[/tex]
