To determine which of the given expressions produces a quadratic function, we need to analyze each expression step-by-step.
Given functions:
[tex]\[ a(x) = 2x - 4 \][/tex]
[tex]\[ b(x) = x + 2 \][/tex]
Let's consider each of the provided expressions:
1. [tex]\((ab)(x)\)[/tex]
- This expression represents the product of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
[tex]\[
(ab)(x) = a(x) \cdot b(x) = (2x - 4)(x + 2)
\][/tex]
- Expanding this product:
[tex]\[
(2x - 4)(x + 2) = 2x \cdot x + 2x \cdot 2 - 4 \cdot x - 4 \cdot 2 = 2x^2 + 4x - 4x - 8 = 2x^2 - 8
\][/tex]
- The result is [tex]\(2x^2 - 8\)[/tex], which is a quadratic function.
2. [tex]\(\left(\frac{a}{b}\right)(x)\)[/tex]
- This expression represents the quotient of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
[tex]\[
\left(\frac{a}{b}\right)(x) = \frac{a(x)}{b(x)} = \frac{2x-4}{x+2}
\][/tex]
- The quotient [tex]\(\frac{2x-4}{x+2}\)[/tex] is not a quadratic function, it is a rational function.
3. [tex]\((a - b)(x)\)[/tex]
- This expression represents the difference between [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
[tex]\[
(a - b)(x) = a(x) - b(x) = (2x - 4) - (x + 2)
\][/tex]
- Simplifying this difference:
[tex]\[
2x - 4 - x - 2 = x - 6
\][/tex]
- The result is [tex]\(x - 6\)[/tex], which is a linear function, not a quadratic function.
4. [tex]\((a + b)(x)\)[/tex]
- This expression represents the sum of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex]:
[tex]\[
(a + b)(x) = a(x) + b(x) = (2x - 4) + (x + 2)
\][/tex]
- Simplifying this sum:
[tex]\[
2x - 4 + x + 2 = 3x - 2
\][/tex]
- The result is [tex]\(3x - 2\)[/tex], which is a linear function, not a quadratic function.
Out of the given expressions, only [tex]\((ab)(x)\)[/tex] results in a quadratic function. Therefore, the expression that produces a quadratic function is:
[tex]\[ \boxed{(ab)(x)} \][/tex]
