91Ó°ÊÓ

When we talk about **function evaluation**, we are referring to the process of finding the output of a function given a specific input. In this context, we're looking at a function \( f(x) = \frac{2}{x} - x^2 \). To evaluate the function, we substitute a value for \( x \) into the expression and compute the result. For instance, if we want to evaluate the function at \( x = 4 \), we substitute 4 in place of \( x \) to calculate \( f(4) = \frac{2}{4} - 4^2 \).

• First, the term \( \frac{2}{4} \) simplifies to \( \frac{1}{2} \).
• Next, \( 4^2 \) equals 16.
• Subtracting these gives \( \frac{1}{2} - 16 = -\frac{31}{2} \).

So, for \( x=4 \), the function evaluates to \( f(4) = -\frac{31}{2} \). If the input value is not within the domain, such as \( x = 0 \) or \( x = 1 \) in this case, the function is not defined, and no evaluation can be done for those inputs.

A **function definition** provides a rule that relates each input to exactly one output. In this case, the rule is given by the expression \( f(x) = \frac{2}{x} - x^2 \). This rule tells us how to transform any input \( x \) to produce the corresponding output. Each function can be represented in several ways: analytically through an equation, graphically, or in words.

• Analytical Representation: The formula \( f(x) = \frac{2}{x} - x^2 \) gives a clear algebraic description of the function.
• Graphical Representation: The graph of this function would show all the input-output pairs, illustrating how the output changes as \( x \) varies.
• Word Representation: This can be described by stating "The output is two divided by the input minus the input squared."

Understanding a function's definition is crucial as it informs us how to perform evaluation and what kind of solutions or restrictions might exist.

The domain of a function is the set of all possible input values \( x \) for which the function is defined. **Domain restrictions** are those values which cannot be included in the domain. For the function \( f(x) = \frac{2}{x} - x^2 \), the given domain is \([2, +\infty)\). This means:

• \( x \) must be greater than or equal to 2.
• The function is not defined for \( x < 2 \) within this specified domain.

This restriction exists because the rule of the function, particularly \( \frac{2}{x} \), might result in undefined behavior when \( x \) is very small (like division by zero for \( x = 0 \), though in this case the domain starts at 2).
The domain ensures that all function evaluations are meaningful and non-problematic, and any calculation outside this range would not adhere to the function's definition within its domain boundary. Being familiar with domains helps in avoiding actions with the function that are mathematically invalid.