Q.75 - Intermediate Algebra

Question

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

$ⓓ (f - g)(-3)$

$f \left(x\right) = 3x^3 - x^2 - 2x + 3$ and $g\left(x\right) = 3x^3- 7x$

Step-by-Step Solution

Answer

The value of $(f+g)\left(x\right)= 6x^3- x^2-9x+3$

The value of $(f+g)\left(2\right)=29$

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

The value of $(f-g)(-3)=-21$

Step 1. Given Information

The given functions are $f \left(x\right) = 3x^3 - x^2 - 2x + 3$and $g\left(x\right) = 3x^3- 7x$

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 )   =   3 x 3   −   x 2   −   2 x   +   3 and g ( x )   =   3 x 3 −   7 x

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 )   =   3 x 3   −   x 2   −   2 x   +   3 and g ( x )   =   3 x 3 −   7 x . The formula is ( f + g ) ( x ) = f ( x ) + g ( x ) Substituting the value in the formula.

$(f+g)\left(x\right)= (3x^3- x^2-2x+3)+( 3x^3- 7x)$

Rewrite without the parentheses

$(f+g)\left(x\right)= 3x^3- x^2-2x+3+ 3x^3- 7x$

Put like terms together

$(f+g)\left(x\right)= 3x^3+ 3x^3- x^2-2x- 7x+3$

Combine like terms

$(f+g)\left(x\right)= 6x^3- x^2-9x+3$

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 ) =   6 x 3 −   x 2 - 9 x + 3 Putting x = 2 in the value of ( f + g ) ( x )

$(f+g)\left(x\right)= 6x^3- x^2-9x+3$

put $x=2$

$$\begin{gathered} (f+g)\left(2\right)= 6{\left(2\right)}^3- {\left(2\right)}^2-9·2+3 \\ (f+g)\left(2\right)= 6·8-4-9·2+3 \\ (f+g)\left(2\right)= 48-4-18+3 \\ (f+g)\left(2\right)=29 \end{gathered}$$

Part (c) Step 1. The given functions are f   ( x )   =   3 x 3   −   x 2   −   2 x   +   3 and g ( x )   =   3 x 3 −   7 x

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 )   =   3 x 3   −   x 2   −   2 x   +   3 and g ( x )   =   3 x 3 −   7 x The formula is ( f - g ) ( x ) = f ( x ) - g ( x ) Substituting the value in the formula.

$(f-g)\left(x\right)= (3x^3- x^2-2x+3)-( 3x^3- 7x)$

Rewrite without the parentheses

$(f-g)\left(x\right)= 3x^3- x^2-2x+3-3x^3+7x$

Put like terms together

$(f-g)\left(x\right)= 3x^3-3x^3- x^2-2x+7x+3$

Combine like terms

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

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 ) = −   x 2 + 5 x + 3 Putting x = - 3 in the value of ( f - g ) ( x )

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

put $x=-3$

$$\begin{gathered} (f-g)(-3)=-{(-3)}^2+5(-3)+3 \\ (f-g)(-3)=-9-15+3 \\ (f-g)(-3)=-21 \end{gathered}$$