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Function evaluation is essentially calculating the output of a function given specific input values. In the problem, we have a specific function, \( h(x) = \frac{x^2 - 1}{x^2 + 1} \), a rational function. To evaluate this function for a given \( x \) value, simply replace \( x \) with the number in the expression.

For example, to find \( h(0.5) \), substitute 0.5 into the function, resulting in \( h(0.5) = \frac{(0.5)^2 - 1}{(0.5)^2 + 1} \). Follow similar steps for each value of \( x \) like 1.5, 2.5, up to 10.5. This process helps us understand how the function behaves at different points.

• Identify the function and the given \( x \) values.
• Substitute the \( x \) values into the function.
• Simplify to find the output—here rounded to four decimal places.

This is a fundamental concept in calculus and algebra that helps in analyzing how functions operate across various domains and is applicable to other complex mathematical tasks.

Rational functions are fractions where both the numerator and the denominator are polynomials. In this exercise, the function \( h(x) = \frac{x^2 - 1}{x^2 + 1} \) is a rational function. The behavior of rational functions can vary critically with changes in \( x \) values, and understanding them is essential for grasping calculus concepts.

A crucial aspect is to recognize that the function is undefined whenever the denominator equals zero; however, this is not a concern here as \( x^2 + 1 \) is always positive for real \( x \). Another point of interest with rational functions is how they asymptotically approach certain y-values as \( x \) increases or decreases.

• Numerator: A polynomial \( x^2 - 1 \)
• Denominator: A polynomial \( x^2 + 1 \)
• No undefined points in real numbers as \( x^2 + 1 \gt 0 \)

Analyzing these elements provides insights into the nature of the function's graph and aids in understanding the roots, asymptotes, and other significant characteristics of the function.

In calculus, intervals allow us to evaluate functions at a sequence of specific points to understand their behavior over a range. Here, we are given an interval from 0.5 to 10.5, increasing in steps of 1, as \( x = 0.5, 1.5, 2.5, \ldots, 10.5 \).

To compute the function over this interval, plug each \( x \) value into the function \( h(x) \) and calculate the result. This practice helps in visualizing the function behavior and identifying trends, such as increases, decreases, and leveling off.

• Start at \( x = 0.5 \) and determine \( h(0.5) \).
• Continue to \( x = 10.5 \) in increments of 1.
• Observe the computed values: they can hint at how the function behaves holistically.

Evaluating these values provides a comprehensive viewpoint on how the rational function changes over this range, which is a crucial aspect of understanding its calculus properties such as limits and continuity.