Question

For functions$f\left(x\right)=2x^2-4x+3$and $g\left(x\right)=x^2-2x-6$ find:$\left(a\right)(f+g)\left(x\right)\left(b\right)(f+g)\left(3\right)\left(c\right)(f-g)\left(x\right)\left(d\right)(f-g)(-2)$

Short answer

The value is

$\begin{array}{l}\left(a\right)(f+g)\left(x\right)=3x^2-6x-3\\ \left(b\right)(f+g)\left(3\right)=6\\ \left(c\right)(f-g)\left(x\right)=x^2-2x+9\\ \left(d\right)(f-g)(-2)=17\end{array}$

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thegivenfunctionsare}\\ f\left(x\right)=2x^2-4x+3\\ g\left(x\right)=x^2-2x-6\end{array}$

We have to find

$\begin{array}{l}\left(a\right)(f+g)\left(x\right)\\ \left(b\right)(f+g)\left(3\right)\\ \left(c\right)(f-g)\left(x\right)\\ \left(d\right)(f-g)(-2)\end{array}$

Step 2. Solve for part (a)

$\begin{array}{l}\text{Weknow}(f+g)\left(x\right)=f\left(x\right)+g\left(x\right)\\ \text{Substitutethefunctions}f\left(x\right)\text{and}g\left(x\right)\\ \Rightarrow (f+g)\left(x\right)=\left(2x^2-4x+3\right)+\left(x^2-2x-6\right)\\ =2x^2-4x+3+x^2-2x-6\left[\text{Rewritewithoutparentheses}\right]\\ =2x^2+x^2-4x-2x+3-6\left[\text{Putliketermstogether}\right]\\ =3x^2-6x-3\end{array}$

Step 3. Solve for part (b)

$\begin{array}{l}\text{Wehave}(f+g)\left(x\right)=3x^2-6x-3\\ \Rightarrow (f+g)\left(3\right)=3{\left(3\right)}^2-6\left(3\right)-3\left[\text{Substitute}x=3\right]\\ =3\left(9\right)-18-3\\ =27-18-3\\ =6\end{array}$

Step 4. Solve for part (c)

$\begin{array}{l}\text{Weknow}(f-g)\left(x\right)=f\left(x\right)-g\left(x\right)\\ \text{Substitutethefunctions}f\left(x\right)\text{and}g\left(x\right)\\ \Rightarrow (f-g)\left(x\right)=\left(2x^2-4x+3\right)-\left(x^2-2x-6\right)\\ =2x^2-4x+3-x^2+2x+6\left[\text{Rewritewithoutparentheses}\right]\\ =2x^2-x^2-4x+2x+3+6\left[\text{Putliketermstogether}\right]\\ =x^2-2x+9\end{array}$

Step 5. Solve for part (d)

$\begin{array}{l}\text{Wehave}(f-g)\left(x\right)=x^2-2x+9\\ \Rightarrow (f-g)(-2)={\left(-2\right)}^2-2\left(-2\right)+9\left[\text{Substitute}x=-2\right]\\ =4+4+9\\ =17\end{array}$