- We identify \(a(x) = 2x-4\) and \(b(x) = x+2\) as linear functions.
- We perform each operation:
- \((a+b)(x) = 3x-2\) (linear)
- \((a/b)(x) = \frac{2x-4}{x+2}\) (rational)
- \((ab)(x) = (2x-4)(x+2) = 2x^2-8\) (quadratic)
- \((a-b)(x) = x-6\) (linear)
- Only the product \((ab)(x)\) results in a polynomial of degree 2.
- The final answer is \(\boxed{c}\).
Explanation
- Analysis of the Functions and Quadratic Form
We are given two linear functions, \(a(x) = 2x - 4\) and \(b(x) = x + 2\). A quadratic function is a polynomial of the form \(f(x) = Ax^2 + Bx + C\), where \(A, B,\) and \(C\) are constants and \(A
eq 0\). We need to determine which of the provided operations result in a quadratic function.
- Evaluating Each Operation
We evaluate each of the given operations individually:
A\((a + b)(x) = a(x) + b(x) = (2x - 4) + (x + 2) = 3x - 2\). This is a linear function of degree 1.
B\(\left(\frac{a}{b}\right)(x) = \frac{a(x)}{b(x)} = \frac{2x - 4}{x + 2}\). This is a rational function, not a polynomial of degree 2.
C\((ab)(x) = a(x) \cdot b(x) = (2x - 4)(x + 2)\). Expanding this expression:
\(2x(x) + 2x(2) - 4(x) - 4(2) = 2x^2 + 4x - 4x - 8 = 2x^2 - 8\). This is a polynomial of degree 2 with \(A=2, B=0, C=-8\).
D\((a - b)(x) = a(x) - b(x) = (2x - 4) - (x + 2) = 2x - 4 - x - 2 = x - 6\). This is a linear function of degree 1.
- Identifying the Quadratic Function and Final Conclusion
Comparing our results to the definition of a quadratic function, only operation (c) produces a function of the form \(Ax^2 + Bx + C\) with \(A
eq 0\). Specifically, \((ab)(x) = 2x^2 - 8\). Therefore, the correct expression is (c).
Examples
In real life, quadratic functions are essential for modeling physical trajectories. For example, if you throw a ball, its height over time is often modeled by a quadratic function. The product of two linear components, like the force and the time it is applied (under certain conditions), can create a quadratic relationship, just as the multiplication of the two linear functions \(a(x)\) and \(b(x)\) resulted in a quadratic function in our problem. This shows how simple building blocks can lead to more complex and useful mathematical models.