[tex](ab)(x)[/tex] Produces quadratic function
Solution:
Given that
a (x) = 2x – 4 and b(x) = x + 2
Need to check which of the expression from given three expression produce a quadratic function. Let us solve each option and check the result.
[tex](a b)(x)=a(x) \times b(x)=(2 x-4) \times(x+2)[/tex]
[tex]\begin{array}{l}{=x(2 x-4)+2(2 x-4)} \\\\ {=2 x^{2}-4 x+4 x-8} \\\\ {=2 x^{2}-8}\end{array}[/tex]
[tex]\Rightarrow \quad(a b)(x)=2 x^{2}-8[/tex]
So (ab)(x) produces a quadratic function [tex]2x^2-8.[/tex]
[tex]\text { 2) }(a-b)(x)=a(x)-b(x)=(2 x-4)-(x+2)[/tex]
[tex]\begin{array}{l}{=(2 x-4)-(x+2)} \\\\ {=2 x-4-x-2} \\\\ {=x-6}\end{array}[/tex]
[tex]\Rightarrow(a-b)(x)=x-6[/tex]
So (a - b)(x) produces a linear function x – 6.
[tex]\begin{array}{l}{\text { 3) }(a+b)(x)=a(x)+b(x)=(2 x-4)+(x+2)} \\\\ {=(2 x-4)+(x+2)} \\\\ {=2 x-4+x+2} \\\\ {=x-2}\end{array}[/tex]
[tex]\Rightarrow(a+b)(x)=x-2[/tex]
So (a + b)(x) produces a linear function x – 2.
Hence we can conclude that (ab)(x) produces quadratic function.
