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Sections 1.1 thru 1.4)
Use analytical and/or graphical methods to determine the intervals on

f(x)=(1)/(x+3)

Sections $1$ . $1$ thru $1$ . $4$ ) $\newline$ Use analytical and/or graphical methods to determine the intervals on $\newline$ $f(x)=\frac{1}{x+3}$

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Full solution

Q. Sections

1

.

1

thru

1

.

4

)

\newline

Use analytical and/or graphical methods to determine the intervals on

\newline

f(x)=\frac{1}{x+3}

Identify Function & Domain: Identify the function and its domain. $f(x) = \frac{1}{(x+3)}$ . The domain of $f(x)$ excludes $x = -3$ , where the function is undefined.
Find Derivative: Find the derivative $f'(x)$ to analyze the behavior of $f(x)$ . Using the quotient rule or the derivative of inverse functions, $f'(x) = -\frac{1}{(x+3)^2}$ .
Determine Sign: Determine the sign of $f'(x)$ to find increasing or decreasing intervals. $\newline$ Since the denominator $(x+3)^2$ is always positive, $f'(x) = -\frac{1}{(x+3)^2}$ is always negative except at $x = -3$ where it's undefined.
Conclude Behavior: Conclude the behavior of $f(x)$ based on the sign of $f'(x)$ . $\newline$ Since $f'(x)$ is always negative, $f(x)$ is always decreasing except at $x = -3$ where it does not exist.

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