Question

Suppose that the functions rand s are defined for all real numbers x as follows

Answer 1

`\begin{gathered} (a)(s+r)(x)=x^2+2x^3 \\ (b)(s\cdot r)(x)=2x^5 \\ (c)(s-r)(2)=-12 \end{gathered}`

Explanation:

Given the functions r and s:

`\begin{gathered} r(x)=2x^3 \\ s(x)=x^2 \end{gathered}`

(a) (s+r)(x)

`\begin{gathered} (s+r)(x)=s(x)+r(x) \\ \implies(s+r)(x)=x^2+2x^3 \end{gathered}`

(b) (s r)(x)

`\begin{gathered} (s\cdot r)(x)=s(x)* r(x) \\ =x^2*2x^3 \\ =2* x^2* x^3 \\ =2* x^(2+3) \\ =2* x^5 \\ \implies(s\cdot r)(x)=2x^5 \end{gathered}`

(c) (s-r)(2)

`\begin{gathered} (s-r)(x)=s(x)-r(x) \\ \implies(s-r)(x)=x^2-2x^3 \end{gathered}`

To find (s-r)(2), substitute 2 for x:

`\begin{gathered} (s-r)(2)=2^2-2(2)^3 \\ =4-2(8) \\ =4-16 \\ =-12 \end{gathered}`

The value of (s-r)(2) is -12.