Domain: All real x, except x=-2 and x=2 f(-3)=-1, f(0)=0, f(3)=1 f^'(x)<0 on (-∞,-2) and (2, ∞)

step 1 So there's a lot of properties we want to satisfy, such as negative 3, negative 1, 0 ,0, and 3,

Domain: All real x, except x=-2 and x=2 f(-3)=-1, f(0)=0,...

Domain: All real $x,$ except $x=-2$ and $x=2$ $f(-3)=-1, f(0)=0, f(3)=1$ $f^{\prime}(x)<0$ on $(-\infty,-2)$ and $(2, \infty)$ $f^{\prime}(x)>0$ on (-2,2) $f^{\prime \prime}(x)<0$ on $(-\infty,-2)$ and (-2,0) $f^{\prime \prime}(x)>0$ on (0,2) and $(2, \infty)$ vertical asymptotes: $x=-2$ and $x=2$ horizontal asymptote: $y=0$

Domain: All real $x,$ except $x=-2$ and $x=2$
$f(-3)=-1, f(0)=0, f(3)=1$
$f^{\prime}(x)<0$ on $(-\infty,-2)$ and $(2, \infty)$
$f^{\prime}(x)>0$ on (-2,2)
$f^{\prime \prime}(x)<0$ on $(-\infty,-2)$ and (-2,0)
$f^{\prime \prime}(x)>0$ on (0,2) and $(2, \infty)$
vertical asymptotes: $x=-2$ and $x=2$
horizontal asymptote: $y=0$

Key Concepts

Domain

The domain of a function refers to the complete set of input values for which the function is defined. Understanding the domain is essential as it outlines where the function can be used and highlights any restrictions, such as values that cause division by zero or other undefined operations.

Monotonicity

Monotonicity describes whether a function is increasing or decreasing over certain intervals. By examining the sign of the first derivative, f'(x), one can determine where the function is rising (when f'(x) > 0) or falling (when f'(x) < 0). This concept is crucial in analyzing the behavior of functions and identifying local extrema.

Concavity

Concavity indicates the direction in which a function curves and is determined by the second derivative, f''(x). If f''(x) is positive, the function is concave up (shaped like a cup), and if f''(x) is negative, the function is concave down (shaped like a cap). This concept is important for understanding the bending behavior of functions and identifying inflection points where the concavity changes.

Asymptotic Behavior

Asymptotic behavior concerns how a function behaves as the input values become very large or approach critical points where the function is not defined. Vertical asymptotes occur when the function grows without bound near specific x-values, indicating a form of discontinuity, while horizontal asymptotes describe the behavior of the function as x approaches infinity or negative infinity, often representing the long-term trend of the function.

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