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Derivatives are a fundamental concept in calculus that describe how a function changes. Specifically, the first derivative of a function, denoted as \(f'(x)\), tells us the rate at which the function's value is changing at any given point. If \(f'(x) > 0\), the function is increasing, meaning the function moves upwards as \(x\) increases. Conversely, if \(f'(x) < 0\), the function is decreasing, indicating a downward movement.

The second derivative, \(f''(x)\), gives us insight into the curvature or concavity of the function. If \(f''(x) > 0\), the graph is concave up, resembling a cup "U" shape, indicating that the slope of the function is increasing. If \(f''(x) < 0\), the graph is concave down, like an upside-down cup "∩", showing a decreasing slope. In this problem, for \(x < 0\), \(f'(x) > 0\) and \(f''(x) > 0\) imply an increasing function that is concave up. For \(x > 0\), \(f'(x) < 0\) and \(f''(x) < 0\) reveal a decreasing and concave down behavior.

Concavity is about how the curve of a function bends. Think of it as the direction the graph "holds" water. If a graph is concave up (\(f''(x) > 0\)), it can hold water, meaning it curves upwards like a smile. If a graph is concave down (\(f''(x) < 0\)), it curves down like a frown, spilling water. This bending or curvature informs how quickly or slowly a function is increasing or decreasing.

In our specific case, for \(x < 0\), \(f''(x) > 0\) implies that the function not only increases, but the rate of increase is accelerating. This results in a graph that gracefully rises towards the point \((0,0)\). Once \(x > 0\), \(f''(x) < 0\) shows a function that is slowing down its descent, producing a smooth decline and leveling off nearer to its end behavior, approaching the horizontal asymptote at \(y = 1\). Understanding concavity is crucial to sketching functions correctly, as it affects the "shape" of the graph.

Asymptotes are lines that a graph approaches but never quite reaches, serving as indicators of the function's end behavior. They can be horizontal, vertical, or even oblique, depending on the function's characteristics. In this problem, the function has two horizontal asymptotes: \(y = -1\) as \(x \to -\infty\) and \(y = 1\) as \(x \to \infty\).

Horizontal asymptotes suggest that as the function goes far to the left or right, it flattens out to a particular value. For \(x \to -\infty\), \(f(x)\) settles near \(-1\), meaning no matter how far left you go, the function's value gets closer to \(-1\) but never actually becomes \(-1\). Similarly, as \(x \to \infty\), the function's value approaches \(1\), leveling off into this asymptote. Understanding asymptotes helps anticipate the long-run behavior of functions as they approach infinitely large or small values.

Graphing functions can seem tricky, but it's all about understanding the function's behavior in different regions. Using derivatives and concavity information, graphs can be sketched to depict the accurate motion and curvature of the function.

In our exercise, for \(x < 0\), you start the graph slightly below \(y = -1\) and curve it upward toward \( (0,0) \), making sure it appears steep near zero and smoothing into the asymptote as \(x \to -\infty\). For \(x > 0\), begin at \( (0,0) \) and draw the curve downward, flattening toward \(y = 1\) as \(x \to \infty\).

The graph should be continuous and smooth, especially at the origin where both parts meet. Combining these insights results in a seamless graph that represents the given mathematical conditions properly. Visualizing and plotting each section's behavior helps create an accurate graph that satisfies all the requirements from increasing, decreasing, and concavity to asymptotic behavior.