Q.76 - Intermediate Algebra

Question

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

$ⓓ (f - g)(-3)$

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

Step-by-Step Solution

Answer

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

The value of $(f+g)\left(2\right)=109$

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

The value of $(f-g)(-3)=68$

Step 1. Given Information

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. 

Part (a) Step 1. The given functions are f ( x ) =   5 x 3 − x 2 + 3 x + 4 and g ( x ) = 8 x 3 − 1

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

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

Part (a) Step 2. We know f ( x ) =   5 x 3 − x 2 + 3 x + 4 and g ( x ) = 8 x 3 − 1 . The formula is ( f + g ) ( x ) = f ( x ) + g ( x ) Substituting the value in the formula.

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

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

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

Part (b) Step 1. We have to find the value of ( f + g ) ( 2 )

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 $(f+g)\left(x\right)$.

Part (b) Step 2. The value of ( f + g ) ( x ) = 13 x 3 − x 2 + 3 x + 3 Putting x = 2 in the value of ( f + g ) ( x )

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

put $x=2$

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

Part (c) Step 1. The given functions are f ( x ) =   5 x 3 − x 2 + 3 x + 4 and g ( x ) = 8 x 3 − 1

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

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

Part (c) Step 2. We know f ( x ) =   5 x 3 − x 2 + 3 x + 4 and g ( x ) = 8 x 3 − 1 The formula is ( f - g ) ( x ) = f ( x ) - g ( x ) Substituting the value in the formula.

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

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

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

Part (d) Step 1. We have to find the value of ( f - g ) ( - 3 )

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 $(f-g)\left(x\right)$

Part (d) Step 2. The value of ( f - g ) ( x ) = - 3 x 3 − x 2 + 3 x + 5 Putting x = - 3 in the value of ( f - g ) ( x )

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

put $x=-3$

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