For each function, find$ⓐ(f+g)\left(x\right)ⓑ(f+g)\left(2\right)ⓒ(f-g)\left(x\right)$

$ⓓ(f-g)(-3)$

localid="1644586522839" $f\left(x\right)=5x^3-x^2+3x+4$andlocalid="1644587634874" $g\left(x\right)=8x^3-1$

The given functions are$f\left(x\right)=5x^3-x^2+3x+4$and$g\left(x\right)=8x^3-1$

We find $ⓐ(f+g)\left(x\right)ⓑ(f+g)\left(2\right)ⓒ(f-g)\left(x\right)ⓓ(f-g)(-3)$for the given function.

We find the value of $(f+g)\left(x\right)$for the given function.

We know$(f+g)\left(x\right)=f\left(x\right)+g\left(x\right)$

$(f+g)\left(x\right)=(5x^3-x^2+3x+4)+(8x^3-1)$

Rewrite without the parentheses

$(f+g)\left(x\right)=5x^3-x^2+3x+4+8x^3-1$

Put like terms together

$(f+g)\left(x\right)=5x^3+8x^3-x^2+3x+4-1$

Combine like terms

$(f+g)\left(x\right)=13x^3-x^2+3x+3$

In part (a) we found $(f+g)\left(x\right)$and now we have to find $(f+g)\left(2\right)$.

To find $(f+g)\left(2\right)$we substitute $x=2$in the value of $(f+g)\left(x\right)$.

$(f+g)\left(x\right)=13x^3-x^2+3x+3$

put $x=2$

$$\begin{gathered} (f+g)\left(2\right)=13{\left(2\right)}^3-{\left(2\right)}^2+3·2+3 \\ (f+g)\left(2\right)=13·8-4+3·2+3 \\ (f+g)\left(2\right)=104-4+6+3 \\ (f+g)\left(2\right)=109 \end{gathered}$$

We have to find the value of $(f-g)\left(x\right)$for the given function.

We know$(f-g)\left(x\right)=f\left(x\right)-g\left(x\right)$

$(f-g)\left(x\right)=(5x^3-x^2+3x+4)-(8x^3-1)$

Rewrite without the parentheses

$(f-g)\left(x\right)=5x^3-x^2+3x+4-8x^3+1$

Put like terms together

$(f-g)\left(x\right)=5x^3-8x^3-x^2+3x+4+1$

Combine like terms

$(f-g)\left(x\right)=-3x^3-x^2+3x+5$

In part (c) we found $(f-g)\left(x\right)$now we have to find$(f-g)(-3)$

To find the value of $(f-g)(-3)$we put $x=-3$in the value of$(f-g)\left(x\right)$

$(f-g)\left(x\right)=-3x^3-x^2+3x+5$

put $x=-3$

$$\begin{gathered} (f-g)(-3)=-3{(-3)}^3-{(-3)}^2+3(-3)+5 \\ (f-g)(-3)=-3(-27)-9+3(-3)+5 \\ (f-g)(-3)=81-9-9+5 \\ (f-g)(-3)=68 \end{gathered}$$