Question

1. given $f(x) = x^2 + 5x + 6$ and $g(x) = x + 2$, find: $(f + g)(x) = \square$ $(f - g)(x) = \square$

Explanation:

Step1: Find \((f + g)(x)\)

To find \((f + g)(x)\), we use the definition of function addition: \((f + g)(x)=f(x)+g(x)\). Substitute \(f(x)=x^{2}+5x + 6\) and \(g(x)=x + 2\) into the formula:
\[

$$\begin{align*} (f + g)(x)&=(x^{2}+5x + 6)+(x + 2)\\ &=x^{2}+(5x+x)+(6 + 2)\\ &=x^{2}+6x+8 \end{align*}$$

\]

Step2: Find \((f - g)(x)\)

To find \((f - g)(x)\), we use the definition of function subtraction: \((f - g)(x)=f(x)-g(x)\). Substitute \(f(x)=x^{2}+5x + 6\) and \(g(x)=x + 2\) into the formula:
\[

$$\begin{align*} (f - g)(x)&=(x^{2}+5x + 6)-(x + 2)\\ &=x^{2}+5x + 6-x - 2\\ &=x^{2}+(5x-x)+(6 - 2)\\ &=x^{2}+4x + 4 \end{align*}$$

\]

Answer:

\((f + g)(x)=\boldsymbol{x^{2}+6x + 8}\)
\((f - g)(x)=\boldsymbol{x^{2}+4x + 4}\)