Question

Question below

Suppose that the functions
r

and
s

are defined for all real numbers
x

as follows.
=
r
x
−
x
4


=
s
x
+
2
x
2

Write the expressions for
+
r
s
x

and
·
r
s
x

and evaluate
−
r
s
2

.

Answer 1

`(r+s)(x)=r(x)+s(x)\\\\(r\cdot s)(x)=r(x)\cdot s(x)\\\\(r-s)(x)=r(x)-s(x)\\\\r(x)=x-4,\ s(x)=2x+2.\ \text{Substitute:}\\\\(r+s)(x)=x-4+2x+2=(x+2x)+(-4+2)=\boxed{3x-2}\\\\(r\cdot s)(x)=(x-4)(2x+2)=(x)(2x)+(x)(2)+(-4)(2x)+(-4)(2)\\\\=2x^2+2x-8x-8=\boxed{2x^2-6x-8}\\\\(r-s)(x)=x-4-(2x+2)=x-4-2x-2=(x-2x)+(-4-2)\\\\=\boxed{-x-6}`