Q.73. - Intermediate Algebra

Question

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

$ⓓ (f - g)(-3)$.

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

Step-by-Step Solution

Answer

The value of $(f+g)\left(x\right)=7x^2+4x+4$.

The value of $(f+g)\left(2\right)=40$.

The value of $(f-g)\left(x\right) =-3x^2 -12x -2$

The value of $(f-g)(-3) =7$

Step 1. Given Information

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

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 )   =   2 x 2   −   4 x   +   1 and g ( x )   =   5 x 2   +   8 x   +   3

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 ) = 2 x 2 - 4 x + 1 and g ( x ) = 5 x 2 + 8 x + 3 . The formula is ( f + g ) ( x ) = f ( x ) + g ( x ) Substituting the value in the formula.

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

Rewrite without the parentheses

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

Put like terms together

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

Combine like terms

$(f+g)\left(x\right)=7x^2+4x+4$

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 ) = 7 x 2 + 4 x + 4 Putting x = 2 in the value of ( f + g ) ( x )

$(f+g)\left(x\right)=7x^2+4x+4$

put $x=2$

$$\begin{gathered} (f+g)\left(2\right)=7{\left(2\right)}^2+4·2+4 \\ (f+g)\left(2\right)=7·4+4·2+4 \\ (f+g)\left(2\right)=28+8+4 \\ (f+g)\left(2\right)=40 \end{gathered}$$

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

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

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

Rewrite without the parentheses

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

Put like terms together

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

Combine like terms

$(f-g)\left(x\right) =-3x^2 -12x -2$

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 2   - 12 x   - 2 Putting x = - 3 in the value of ( f - g ) ( x )  

$(f-g)\left(x\right) =-3x^2 -12x -2$

put $x=-3$

$$\begin{gathered} (f-g)(-3) =-3{(-3)}^2 -12·(-3) -2 \\ (f-g)(-3) =-3·9+12·3 -2 \\ (f-g)(-3) =-27+36 -2 \\ (f-g)(-3) =7 \end{gathered}$$