Question

suppose that the functions r and s are defined for all real numbers x as follows.\\( r(x) = x + 1 \\)\\( s(x) = 2x^2 \\)write the expressions for \\( (r + s)(x) \\) and \\( (r cdot s)(x) \\) and evaluate \\( (r - s)(-3) \\).\\( (r + s)(x) = \\)\\( (r cdot s)(x) = \\)\\( (r - s)(-3) = \\)

Explanation:

Step1: Define sum of functions

$(r+s)(x) = r(x) + s(x)$

Step2: Substitute given functions

$(r+s)(x) = (x+1) + 2x^2 = 2x^2 + x + 1$

Step3: Define product of functions

$(r\cdot s)(x) = r(x) \cdot s(x)$

Step4: Multiply the given functions

$(r\cdot s)(x) = (x+1) \cdot 2x^2 = 2x^3 + 2x^2$

Step5: Define difference of functions

$(r-s)(x) = r(x) - s(x)$

Step6: Substitute $x=-3$

$(r-s)(-3) = r(-3) - s(-3) = [(-3)+1] - [2(-3)^2]$

Step7: Simplify the expression

$(r-s)(-3) = (-2) - (2\cdot9) = -2 - 18 = -20$

Answer:

$(r+s)(x) = 2x^2 + x + 1$
$(r\cdot s)(x) = 2x^3 + 2x^2$
$(r-s)(-3) = -20$