Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To identify which of the given expressions produces a quadratic function when [tex]\(a(x) = 2x - 4\)[/tex] and [tex]\(b(x) = x + 2\)[/tex], let's evaluate each expression step by step.
1. Product of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\((ab)(x)\)[/tex]
[tex]\[ (ab)(x) = a(x) \cdot b(x) \][/tex]
Substituting the given functions:
[tex]\[ (ab)(x) = (2x - 4) \cdot (x + 2) \][/tex]
Expanding using the distributive property:
[tex]\[ (ab)(x) = 2x(x + 2) - 4(x + 2) \][/tex]
[tex]\[ = 2x^2 + 4x - 4x - 8 \][/tex]
Simplifying:
[tex]\[ (ab)(x) = 2x^2 - 8 \][/tex]
This is a quadratic function because it includes an [tex]\(x^2\)[/tex] term.
2. Quotient of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\(\left(\frac{a}{b}\right)(x)\)[/tex]
[tex]\[ \left(\frac{a}{b}\right)(x) = \frac{a(x)}{b(x)} \][/tex]
Substituting the given functions:
[tex]\[ \left(\frac{a}{b}\right)(x) = \frac{2x - 4}{x + 2} \][/tex]
This is a rational function, not a quadratic function because it does not have an [tex]\(x^2\)[/tex] term as a polynomial.
3. Difference of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\((a - b)(x)\)[/tex]
[tex]\[ (a - b)(x) = a(x) - b(x) \][/tex]
Substituting the given functions:
[tex]\[ (a - b)(x) = (2x - 4) - (x + 2) \][/tex]
Simplifying:
[tex]\[ (a - b)(x) = 2x - 4 - x - 2 \][/tex]
[tex]\[ = x - 6 \][/tex]
This is a linear function because it does not have an [tex]\(x^2\)[/tex] term.
4. Sum of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\((a + b)(x)\)[/tex]
[tex]\[ (a + b)(x) = a(x) + b(x) \][/tex]
Substituting the given functions:
[tex]\[ (a + b)(x) = (2x - 4) + (x + 2) \][/tex]
Simplifying:
[tex]\[ (a + b)(x) = 2x - 4 + x + 2 \][/tex]
[tex]\[ = 3x - 2 \][/tex]
This is a linear function because it does not have an [tex]\(x^2\)[/tex] term.
Therefore, the only expression that produces a quadratic function is [tex]\((ab)(x) = 2x^2 - 8\)[/tex]. Hence:
[tex]\((ab)(x)\)[/tex] is the expression that produces a quadratic function.
1. Product of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\((ab)(x)\)[/tex]
[tex]\[ (ab)(x) = a(x) \cdot b(x) \][/tex]
Substituting the given functions:
[tex]\[ (ab)(x) = (2x - 4) \cdot (x + 2) \][/tex]
Expanding using the distributive property:
[tex]\[ (ab)(x) = 2x(x + 2) - 4(x + 2) \][/tex]
[tex]\[ = 2x^2 + 4x - 4x - 8 \][/tex]
Simplifying:
[tex]\[ (ab)(x) = 2x^2 - 8 \][/tex]
This is a quadratic function because it includes an [tex]\(x^2\)[/tex] term.
2. Quotient of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\(\left(\frac{a}{b}\right)(x)\)[/tex]
[tex]\[ \left(\frac{a}{b}\right)(x) = \frac{a(x)}{b(x)} \][/tex]
Substituting the given functions:
[tex]\[ \left(\frac{a}{b}\right)(x) = \frac{2x - 4}{x + 2} \][/tex]
This is a rational function, not a quadratic function because it does not have an [tex]\(x^2\)[/tex] term as a polynomial.
3. Difference of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\((a - b)(x)\)[/tex]
[tex]\[ (a - b)(x) = a(x) - b(x) \][/tex]
Substituting the given functions:
[tex]\[ (a - b)(x) = (2x - 4) - (x + 2) \][/tex]
Simplifying:
[tex]\[ (a - b)(x) = 2x - 4 - x - 2 \][/tex]
[tex]\[ = x - 6 \][/tex]
This is a linear function because it does not have an [tex]\(x^2\)[/tex] term.
4. Sum of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]: [tex]\((a + b)(x)\)[/tex]
[tex]\[ (a + b)(x) = a(x) + b(x) \][/tex]
Substituting the given functions:
[tex]\[ (a + b)(x) = (2x - 4) + (x + 2) \][/tex]
Simplifying:
[tex]\[ (a + b)(x) = 2x - 4 + x + 2 \][/tex]
[tex]\[ = 3x - 2 \][/tex]
This is a linear function because it does not have an [tex]\(x^2\)[/tex] term.
Therefore, the only expression that produces a quadratic function is [tex]\((ab)(x) = 2x^2 - 8\)[/tex]. Hence:
[tex]\((ab)(x)\)[/tex] is the expression that produces a quadratic function.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
