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

step 1 So we have a lot of properties to maintain here. We want the domain to be all real numbers excep

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

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

Key Concepts

Domain and Continuity

The domain of a function specifies all input values for which the function is defined, including any restrictions such as points of discontinuity or excluded values. In problems like this, recognizing the domain is essential because it informs where the function can be analyzed, and where any discontinuities, such as vertical asymptotes, might occur.

Monotonicity and the First Derivative

The first derivative of a function provides information about the increasing or decreasing nature (monotonicity) of that function. If the derivative is positive over an interval, the function is strictly increasing on that interval, which is crucial for determining how the function behaves between given points.

Concavity and the Second Derivative

The second derivative indicates the concavity of a function, which describes whether the graph curves upward (convex) or downward (concave). If the second derivative is positive on an interval, the function is concave up, and if it is negative, the function is concave down. This helps in understanding the curvature and locating potential points of inflection.

Vertical Asymptote

A vertical asymptote occurs when a function approaches infinity or negative infinity as the input approaches a particular value, indicating a discontinuity in the function. Recognizing vertical asymptotes is important for understanding the behavior of a function near points where it is not defined.

Horizontal Asymptote

A horizontal asymptote describes the behavior of a function as the input values become very large in the positive or negative direction. When the output of a function approaches a constant value, the horizontal asymptote provides insight into the long-term behavior of the function.

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