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

step 1 So for this problem, we're going to have a couple of important values. We have the points 0, neg

Domain: All real x, except x=1 ; f(0)=-2, f(2)=0 f^'(x)<0 on...

Domain: All real $x,$ except $x=1 ;$
$f(0)=-2, f(2)=0$
$f^{\prime}(x)<0$ 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

Monotonicity

Monotonicity refers to the quality of a function to be exclusively either increasing or decreasing over an interval. When the first derivative of a function is strictly negative over an interval, it implies that the function is monotonically decreasing on that interval, meaning that as the input increases, the output consistently decreases.

Domain

The domain of a function is the complete set of all possible input values for which the function is defined. In many problems, sometimes certain values are excluded because they lead to undefined expressions such as division by zero or other discontinuities, which is crucial for understanding the function’s behavior across its entire input spectrum.

Concavity

Concavity describes the curvature of a function's graph and is determined by its second derivative. If the second derivative is negative over an interval, the graph is concave down, indicating that the slope of the function is decreasing, while a positive second derivative suggests that the graph is concave up, meaning the slope increases. This information is vital in understanding the bending behavior and potential inflection points of the graph.

Vertical Asymptote

A vertical asymptote is a line x = a where the function's values grow without bound as the input approaches a. This generally occurs because the function is not defined at that point, often due to a division by zero or other discontinuities, marking a fundamental change in the behavior of the function near the asymptote.

Horizontal Asymptote

A horizontal asymptote is a horizontal line y = b that the graph of the function approaches as the input values tend to infinity or negative infinity. It describes the end-behavior of the function, indicating how the outputs level off, and is an essential aspect of analyzing functions that have a predictable long-term behavior.

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