Q.74 - Intermediate Algebra

Question

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

$ⓓ (f - g)(-3)$

$f \left(x\right) = 4x^2 - 7x + 3$ and $g\left(x\right) = 4x^2 + 2x - 1$

Step-by-Step Solution

Answer

The value of $(f+g)\left(x\right)=8x^2-5x+2$

The value of $(f+g)\left(2\right)=24$

The value of $(f-g)\left(x\right)=-9x+4$

The value of $(f-g)(-3)=31$

Step 1. Given Information

The given functions are $f \left(x\right) = 4x^2 - 7x + 3$ and$g\left(x\right) = 4x^2 + 2x - 1$

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 )   =   4 x 2   −   7 x   +   3 and g ( x )   =   4 x 2   +   2 x   −   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 )   =   4 x 2   −   7 x   +   3 and g ( x )   =   4 x 2   +   2 x   −   1 . The formula is ( f + g ) ( x ) = f ( x ) + g ( x ) Substituting the value in the formula.

$(f+g)\left(x\right)=(4x^2-7x+3)+(4x^2+2x-1)$

Rewrite without the parentheses

$(f+g)\left(x\right)=4x^2-7x+3+4x^2+2x-1$

Put like terms together

$(f+g)\left(x\right)=4x^2+4x^2+2x-7x+3-1$

Combine like terms

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

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 ) = 8 x 2 - 5 x + 2 Putting x = 2 in the value of ( f + g ) ( x )

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

put $x=2$

$$\begin{gathered} (f+g)\left(2\right)=8{\left(2\right)}^2-5·2+2 \\ (f+g)\left(2\right)=8·4-5·2+2 \\ (f+g)\left(2\right)=32-10+2 \\ (f+g)\left(2\right)=24 \end{gathered}$$

Part (c) Step 1. The given functions are f   ( x )   =   4 x 2   −   7 x   +   3 and g ( x )   =   4 x 2   +   2 x   −   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 (c) Step 2. We know f   ( x )   =   4 x 2   −   7 x   +   3 and g ( x )   =   4 x 2   +   2 x   −   1 The formula is ( f - g ) ( x ) = f ( x ) - g ( x ) Substituting the value in the formula.

$(f-g)\left(x\right)=(4x^2-7x+3)-(4x^2+2x-1)$

Rewrite without the parentheses

$(f-g)\left(x\right)=4x^2-7x+3-4x^2-2x+1$

Put like terms together

$(f-g)\left(x\right)=4x^2-4x^2-2x-7x+3+1$

Combine like terms

$(f-g)\left(x\right)=-9x+4$

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 ) = - 9 x + 4 Putting x = - 3 in the value of ( f - g ) ( x )

$(f-g)\left(x\right)=-9x+4$

put $x=-3$

$$\begin{gathered} (f-g)(-3)=-9(-3)+4 \\ (f-g)(-3)=27+4 \\ (f-g)(-3)=31 \end{gathered}$$