if a(x) = 2x - 4 and b(x) = x + 2, which of the following expressions produces a quadratic…

if a(x) = 2x - 4 and b(x) = x + 2, which of the following expressions produces a quadratic function?\n(ab)(x)\n(\\frac{a}{b})(x)\n(a - b)(x)\n(a + b)(x)

if a(x) = 2x - 4 and b(x) = x + 2, which of the following expressions produces a quadratic function?\n(ab)(x)\n(\\frac{a}{b})(x)\n(a - b)(x)\n(a + b)(x)

Answer

Explanation:

Step1: Recall function - multiplication rule

$(ab)(x)=a(x)\times b(x)$.

Step2: Substitute the given functions

Since $a(x) = 2x - 4$ and $b(x)=x + 2$, then $(ab)(x)=(2x - 4)(x + 2)$.

Step3: Expand the product

Using the FOIL method: [ \begin{align*} (2x-4)(x + 2)&=2x\times x+2x\times2-4\times x - 4\times2\ &=2x^{2}+4x-4x - 8\ &=2x^{2}-8 \end{align*} ] which is a quadratic function.

Step4: Check division

$\left(\frac{a}{b}\right)(x)=\frac{2x - 4}{x + 2}$ is a rational - function.

Step5: Check subtraction

$(a - b)(x)=(2x-4)-(x + 2)=2x-4-x - 2=x-6$, which is a linear function.

Step6: Check addition

$(a + b)(x)=(2x-4)+(x + 2)=2x-4+x + 2=3x-2$, which is a linear function.

Answer:

A. $(ab)(x)$