Intermediate Algebra, Chapter 5, 5.3, Section 5.3, Problem 15

If $f(x) = 3x^2 + 9x + 10$ and $g(x) = 4x^2 + 2x + 12$. Find (a) $(f - g)(x)$ (b) $(f + g)(x)$
a.) For $(f + g)(x)$


b.) $(f - g)(x)$

Compose the result function for $f−g$ by replacing the function designators with the actual functions.

$(3x^2+9x+10)−(4x^2+2x+12)$


Multiply $−1$ by each term inside the parentheses.

$3x^2+9x+10−4x^2−2x−12$


Since $3x^2$ and $−4x^2$ are like terms, add $−4x^2$ to $3x^2$ to get $−x^2$.

$−x^2+9x+10−2x−12$


Since $9x$ and $−2x$ are like terms, add $−2x$ to $9x$ to get $7x$.

$−x^2+7x+10−12$


Subtract $12$ from $10$ to get $−2$.

$(f - g)(x) = −x^2+7x−2$

b.) For $(f + g)(x)$
Compose the result function for $f+g$ by replacing the function designators with the actual functions.

$(3x^2+9x+10)+(4x^2+2x+12)$


Remove the parentheses that are not needed from the expression.

$3x^2+9x+10+4x^2+2x+12$


Since $3x^2$ and $4x^2$ are like terms, add $4x^2$ to $3x^2$ to get $7x^2$.

$7x^2+9x+10+2x+12$


Since $9x$ and $2x$ are like terms, add $2x$ to $9x$ to get $11x$.

$7x^2+11x+10+12$


Add $12$ to $10$ to get $22$.

$(f+g)(x) = 7x^2+11x+22$