Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let's go through the steps needed to find \((f \cdot g)(x)\) given the functions:
[tex]\[ f(x) = \sqrt{3x} \][/tex]
[tex]\[ g(x) = \sqrt{48x} \][/tex]
To find \((f \cdot g)(x)\), we need to determine the product of \(f(x)\) and \(g(x)\):
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
First, let's express \(f(x)\) and \(g(x)\) in detail:
[tex]\[ f(x) = \sqrt{3x} \][/tex]
[tex]\[ g(x) = \sqrt{48x} \][/tex]
To find the product, we multiply these two expressions together:
[tex]\[ (f \cdot g)(x) = \sqrt{3x} \cdot \sqrt{48x} \][/tex]
Recall the property of square roots:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Applying this property to our functions:
[tex]\[ (f \cdot g)(x) = \sqrt{3x} \cdot \sqrt{48x} = \sqrt{(3x) \cdot (48x)} \][/tex]
Now multiply the expressions inside the square root:
[tex]\[ (3x) \cdot (48x) = 3 \cdot 48 \cdot x \cdot x = 144x^2 \][/tex]
So, the expression becomes:
[tex]\[ (f \cdot g)(x) = \sqrt{144x^2} \][/tex]
Since \(\sqrt{x^2} = |x|\) and we are given that \(x \geq 0\):
[tex]\[ \sqrt{144x^2} = 12x \][/tex]
Thus, the function \((f \cdot g)(x)\) is:
[tex]\[ (f \cdot g)(x) = 12x \][/tex]
Therefore, the correct answer is:
D. [tex]\((f \cdot g)(x) = 12x\)[/tex]
[tex]\[ f(x) = \sqrt{3x} \][/tex]
[tex]\[ g(x) = \sqrt{48x} \][/tex]
To find \((f \cdot g)(x)\), we need to determine the product of \(f(x)\) and \(g(x)\):
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
First, let's express \(f(x)\) and \(g(x)\) in detail:
[tex]\[ f(x) = \sqrt{3x} \][/tex]
[tex]\[ g(x) = \sqrt{48x} \][/tex]
To find the product, we multiply these two expressions together:
[tex]\[ (f \cdot g)(x) = \sqrt{3x} \cdot \sqrt{48x} \][/tex]
Recall the property of square roots:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Applying this property to our functions:
[tex]\[ (f \cdot g)(x) = \sqrt{3x} \cdot \sqrt{48x} = \sqrt{(3x) \cdot (48x)} \][/tex]
Now multiply the expressions inside the square root:
[tex]\[ (3x) \cdot (48x) = 3 \cdot 48 \cdot x \cdot x = 144x^2 \][/tex]
So, the expression becomes:
[tex]\[ (f \cdot g)(x) = \sqrt{144x^2} \][/tex]
Since \(\sqrt{x^2} = |x|\) and we are given that \(x \geq 0\):
[tex]\[ \sqrt{144x^2} = 12x \][/tex]
Thus, the function \((f \cdot g)(x)\) is:
[tex]\[ (f \cdot g)(x) = 12x \][/tex]
Therefore, the correct answer is:
D. [tex]\((f \cdot g)(x) = 12x\)[/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.