To determine which of the given expressions involving the functions [tex]\(a(x) = 2x - 4\)[/tex] and [tex]\(b(x) = x + 2\)[/tex] results in a quadratic function, we need to evaluate each expression:
1. Product of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
To find [tex]\((a \cdot b)(x)\)[/tex], we multiply [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
[tex]\[
(a \cdot b)(x) = a(x) \cdot b(x) = (2x - 4)(x + 2)
\][/tex]
Use the distributive property to expand:
[tex]\[
(2x - 4)(x + 2) = 2x \cdot x + 2x \cdot 2 - 4 \cdot x - 4 \cdot 2
\][/tex]
Simplify each term:
[tex]\[
= 2x^2 + 4x - 4x - 8
\][/tex]
Combine like terms:
[tex]\[
= 2x^2 - 8
\][/tex]
This is a quadratic function because it contains the term [tex]\(2x^2\)[/tex].
2. Quotient of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
To find [tex]\(\left(\frac{a}{b}\right)(x)\)[/tex], we divide [tex]\(a(x)\)[/tex] by [tex]\(b(x)\)[/tex]:
[tex]\[
\left(\frac{a}{b}\right)(x) = \frac{2x - 4}{x + 2}
\][/tex]
Simplify the expression by factoring the numerator:
[tex]\[
= \frac{2(x - 2)}{x + 2}
\][/tex]
This is a rational function, not a quadratic function.
3. Difference between [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
To find [tex]\((a - b)(x)\)[/tex], we subtract [tex]\(b(x)\)[/tex] from [tex]\(a(x)\)[/tex]:
[tex]\[
(a - b)(x) = a(x) - b(x) = (2x - 4) - (x + 2)
\][/tex]
Distribute the negative sign and combine like terms:
[tex]\[
= 2x - 4 - x - 2 = x - 6
\][/tex]
This is a linear function, not a quadratic function.
4. Sum of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
To find [tex]\((a + b)(x)\)[/tex], we add [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
[tex]\[
(a + b)(x) = a(x) + b(x) = (2x - 4) + (x + 2)
\][/tex]
Combine like terms:
[tex]\[
= 2x + x - 4 + 2 = 3x - 2
\][/tex]
This is a linear function, not a quadratic function.
In conclusion, the only expression that produces a quadratic function is [tex]\((a \cdot b)(x)\)[/tex]. Therefore, the correct answer is:
[tex]\[
(a \cdot b)(x)
\][/tex]
