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.
