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Vertical asymptotes mark the values of \( x \) where a function tends to infinity, and the graph shoots upwards or downwards without bound. These points occur when the function is undefined, typically when the denominator of a rational function equals zero. For the function \( f(x) = \frac{1}{1-x^2} \), the vertical asymptotes are found where the expression \( 1-x^2 = 0 \). By solving for \( x \), the values \( x = \pm 1 \) make the denominator zero, leading to the vertical asymptotes at \( x = 1 \) and \( x = -1 \). These points are where the graph approaches a vertical line, splitting the function's domain into various sections.

Horizontal asymptotes represent the values that a function approaches as \( x \) becomes extremely large or small, specifically as \( x \to \pm \infty \). For rational functions where the degree of the polynomial in the numerator is less than that of the denominator, the horizontal asymptote typically lies at \( y = 0 \). In the case of \( f(x) = \frac{1}{1-x^2} \), there is no polynomial term in the numerator that is of higher degree than the denominator. Thus, as \( x \to \pm \infty \), \( f(x) \) approaches zero, confirming that the horizontal asymptote is \( y = 0 \). This horizontal line indicates that the function will level out at \( y = 0 \) as \( x \) moves towards infinity in either direction.

The concept of concavity helps us understand how the graph of a function curves. A function is *concave up* when it curves upwards like a cup, indicating that the slope of the tangent line is increasing. Conversely, it is *concave down* when it curves downwards like a frown, with the slope of the tangent line decreasing. To find concavity for \( f(x) = \frac{1}{1-x^2} \), we need the second derivative, \( f''(x) \), which tells us about the function's rate of change of slope.

• **Concave Up**: Occurs in the interval \(-1 < x < 1\), where \( f''(x) > 0 \).
• **Concave Down**: Occurs outside the interval, at \( x > 1 \) and \( x < -1 \), where \( f''(x) < 0 \).

The intervals where these occur depend on the sign of the second derivative, leading to distinct sections of the curve's behavior, but some regions of the curve may not show concavity at all.

Monotonicity describes whether a function is consistently increasing or decreasing over an interval. For \( f(x) = \frac{1}{1-x^2} \), finding the first derivative, \( f'(x) \), helps us identify these intervals:

• **Increasing**: Where \( f'(x) > 0 \), meaning the function rises with increasing \( x \). This occurs at \( x > 0 \).
• **Decreasing**: Where \( f'(x) < 0 \), showing the function drops with increasing \( x \). This is found at \( x < 0 \).

In conjunction with concavity, we can identify combinations of behaviors such as concave up and increasing (\(0 < x < 1\)), concave up and decreasing (\(-1 < x < 0\)), and concave down and decreasing (\(x > 1\) and \(x < -1\)). Monotonicity simplifies the task of graphing functions by revealing the function’s tendency over specified intervals.