Question

name: gavin hamilton period: 2 date: 2-3-26 let ( f(x) = x^2 + 2x + 1 ) and ( g(x) = x - 4 ) identify the rule for a. ( f + g ) b. ( f / g )

Explanation:

Part a: \( f + g \)

Step1: Recall the definition of function addition

To find \( (f + g)(x) \), we use the formula \( (f + g)(x)=f(x)+g(x) \).
Given \( f(x)=x^{2}+2x + 1 \) and \( g(x)=x - 4 \), we substitute these into the formula:
\( (f + g)(x)=(x^{2}+2x + 1)+(x - 4) \)

Step2: Combine like terms

First, remove the parentheses: \( x^{2}+2x + 1+x - 4 \)
Then, combine the \( x \)-terms and the constant terms:
The \( x \)-terms are \( 2x+x = 3x \), and the constant terms are \( 1-4=-3 \).
So, \( (f + g)(x)=x^{2}+3x-3 \)

Step1: Recall the definition of function division

To find \( \frac{f}{g}(x) \), we use the formula \( \frac{f}{g}(x)=\frac{f(x)}{g(x)} \), where \( g(x)
eq0 \).
Given \( f(x)=x^{2}+2x + 1 \) and \( g(x)=x - 4 \), we substitute these into the formula:
\( \frac{f}{g}(x)=\frac{x^{2}+2x + 1}{x - 4} \)

Step2: Simplify the numerator (if possible)

Notice that the numerator \( x^{2}+2x + 1 \) is a perfect square trinomial, which factors as \( (x + 1)^{2} \).
So, \( \frac{f}{g}(x)=\frac{(x + 1)^{2}}{x - 4} \), with the restriction \( x
eq4 \) (since the denominator cannot be zero).

Answer:

\( (f + g)(x)=x^{2}+3x - 3 \)

Part b: \( f / g \) (assuming it's \( \frac{f}{g}(x) \))