Question

suppose that the functions r and s are defined for all real numbers x as follows.
r(x)=5x
s(x)=3x+4
write the expressions for (r + s)(x) and (r·s)(x) and evaluate (r - s)(3).
(r + s)(x) =
(r·s)(x) =
(r - s)(3) =

Explanation:

Step1: Define sum of functions

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

Step2: Substitute and simplify sum

$(r+s)(x) = 5x + (3x+4) = 8x + 4$

Step3: Define product of functions

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

Step4: Expand and simplify product

$(r \cdot s)(x) = 5x(3x+4) = 15x^2 + 20x$

Step5: Define difference at $x=3$

$(r-s)(3) = r(3) - s(3)$

Step6: Compute $r(3)$ and $s(3)$

$r(3)=5(3)=15$, $s(3)=3(3)+4=13$

Step7: Calculate the difference

$(r-s)(3) = 15 - 13 = 2$

Answer:

$(r+s)(x) = 8x + 4$
$(r \cdot s)(x) = 15x^2 + 20x$
$(r-s)(3) = 2$