Answered

At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Explore a wealth of knowledge from professionals across various disciplines on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

For which pair of functions is [tex]\((f \circ g)(x) = 12x\)[/tex]?

A. [tex]\(f(x) = 3 - 4x\)[/tex] and [tex]\(g(x) = 16x - 3\)[/tex]

B. [tex]\(f(x) = 6x^2\)[/tex] and [tex]\(g(x) = \frac{2}{x}\)[/tex]

C. [tex]\(f(x) = \sqrt{x}\)[/tex] and [tex]\(g(x) = 144x\)[/tex]

D. [tex]\(f(x) = 4x\)[/tex] and [tex]\(g(x) = 3x\)[/tex]

Sagot :

GinnyAnswer
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].
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.