Question

name hines
5-5 lesson quiz
function operations

1. let ( f(x) = 3x^2 - 2x + 6 ) and ( g(x) = 7x - 4 ). identify the rule for ( f(x) + g(x) ). show work!

a ( 21x^3 - 26x^2 - 50x - 24 )
b ( 21x^2 - 21x + 50 )
c ( 3x^2 + 5x + 2 )
d ( 3x^2 + 9x - 2 )

1. part a

identify the rule for ( \frac{f}{g} ) when ( f(x) = -3x - 6 ) and ( g(x) = x^2 - x - 6 ). show work!
a ( \frac{f(x)}{g(x)} = \frac{-2}{(x + 2)(x - 3)} )
b ( \frac{f(x)}{g(x)} = \frac{3}{(x + 2)} )
c ( \frac{f(x)}{g(x)} = \frac{-3}{x - 3} )
d ( \frac{f(x)}{g(x)} = \frac{-3}{x - 3} )

Explanation:

Problem 1:

Step1: Recall function addition

To find \( f(x) + g(x) \), we add the two functions \( f(x) = 3x^2 - 2x + 6 \) and \( g(x) = 7x - 4 \).
\( f(x)+g(x)=(3x^2 - 2x + 6)+(7x - 4) \)

Step2: Combine like terms

Combine the \( x \)-terms and constant terms:
\( 3x^2+(-2x + 7x)+(6 - 4)=3x^2 + 5x+2 \)

Step1: Recall function division

To find \( \frac{f(x)}{g(x)} \), we substitute \( f(x)=-3x - 6 \) and \( g(x)=x^2 - x - 6 \). First, factor both numerator and denominator.
Numerator: \( -3x - 6=-3(x + 2) \)
Denominator: \( x^2 - x - 6=(x - 3)(x + 2) \) (by factoring \( x^2 - x - 6 \) as \( (x - 3)(x + 2) \) since \( -3\times2=-6 \) and \( -3 + 2=-1 \))

Step2: Simplify the fraction

\( \frac{f(x)}{g(x)}=\frac{-3(x + 2)}{(x - 3)(x + 2)} \). Cancel out the common factor \( (x + 2) \) (assuming \( x
eq - 2 \)) to get \( \frac{-3}{x - 3} \)

Answer:

C. \( 3x^2 + 5x + 2 \)

Problem 2 (Part A):