The problem involves performing function operations on two given functions:
1.
Addition of functions ($f + g$)
2.
Subtraction of functions ($f - g$)
3.
Division of functions ($f / g$)
The given functions are:
- $ f(x) = 2x + 5 $
- $ g(x) = x^2 - 4 $
Step-by-Step Solution:
#### 1.
Addition of Functions ($f + g$)
To find $(f + g)(x)$, we add the expressions for $f(x)$ and $g(x)$:
\[
(f + g)(x) = f(x) + g(x)
\]
Substitute the given functions:
\[
(f + g)(x) = (2x + 5) + (x^2 - 4)
\]
Combine like terms:
\[
(f + g)(x) = x^2 + 2x + 5 - 4
\]
\[
(f + g)(x) = x^2 + 2x + 1
\]
#### 2.
Subtraction of Functions ($f - g$)
To find $(f - g)(x)$, we subtract the expression for $g(x)$ from $f(x)$:
\[
(f - g)(x) = f(x) - g(x)
\]
Substitute the given functions:
\[
(f - g)(x) = (2x + 5) - (x^2 - 4)
\]
Distribute the negative sign:
\[
(f - g)(x) = 2x + 5 - x^2 + 4
\]
Combine like terms:
\[
(f - g)(x) = -x^2 + 2x + 9
\]
#### 3.
Division of Functions ($f / g$)
To find $(f / g)(x)$, we divide the expression for $f(x)$ by $g(x)$:
\[
(f / g)(x) = \frac{f(x)}{g(x)}
\]
Substitute the given functions:
\[
(f / g)(x) = \frac{2x + 5}{x^2 - 4}
\]
Note that the denominator $x^2 - 4$ can be factored as a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]
Thus:
\[
(f / g)(x) = \frac{2x + 5}{(x - 2)(x + 2)}
\]
This is the simplified form, but we must also note the domain restrictions. The function is undefined where the denominator is zero, i.e., $x \neq 2$ and $x \neq -2$.
Final Answers:
\[
\boxed{
\begin{aligned}
(f + g)(x) &= x^2 + 2x + 1 \\
(f - g)(x) &= -x^2 + 2x + 9 \\
(f / g)(x) &= \frac{2x + 5}{(x - 2)(x + 2)}, \quad x \neq 2, x \neq -2
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of function operations algebra 2 worksheet.
