91Ó°ÊÓ

In mathematics, ordered pairs play a crucial role in representing relationships between two sets of numbers. An ordered pair \( (x, y) \) consists of two elements, where the first element \( x \) is known as the abscissa, usually corresponding to the horizontal axis on a graph, and the second element \( y \) is known as the ordinate, usually corresponding to the vertical axis. The order in which the elements appear is important, as the pair \( (x, y) \) is distinct from \( (y, x) \) unless \( x = y \).

In the context of functions, an ordered pair represents an input and its corresponding output, \( (x, f(x)) \). When a function is defined, such as \( f(x) = (x-3)^2 \), and we have a specific domain \( A = \{-2, -1, 0, 1, 2\} \) to work with, we can calculate the output for each input. These results can be paired with their respective inputs to form the set of ordered pairs that define the function in terms of its domain. For instance, with the given function and domain, we calculate each output and then express them as ordered pairs. This process captures the complete function as a set, which in this case is \( \{(-2, 25), (-1, 16), (0, 9), (1, 4), (2, 1)\} \).

Ordered pairs provide a clear, structured way to map out the relationship within functions, making it easier to visualize and understand the function's behavior across different values of \( x \).

The domain of a function is the complete set of possible values of the independent variable \( x \) which will output valid real numbers \( y \) when substituted into the function. In simpler terms, it's the set of all inputs over which the function is defined. It's crucial to establish the domain to prevent undefined or non-real results, such as dividing by zero or taking the square root of a negative number in the realm of real numbers. Considering the domain is particularly important in functions that involve denominators, radicals, and logarithms.

For the given exercise where the function is \( f(x) = (x-3)^2 \) and the explicit domain is given as \( A=\{-2,-1,0,1,2\} \), we focus on just these values. It is important to note that not every function comes with a domain neatly packaged like this. Sometimes, it's necessary to determine the domain by considering the properties of the function itself.

To determine the domain, we often ask: For what values of \( x \) is this function defined? For quadratic functions, like the one in our exercise, the domain is typically all real numbers because you can square any real number. In our case, however, the domain is limited by the set \( A \) provided in the exercise. This restriction allows us to have a finite and specific set of ordered pairs representing the function. Understanding the concept of domain enables students to better understand the scope and limitations of functions in various mathematical contexts.

Quadratic functions are a type of polynomial function with a degree of two, which means the highest exponent of the variable \( x \) is two. The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \) where \( a \) cannot be zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on whether \( a \) is positive or negative.

With quadratic functions, we encounter various properties such as a vertex, which is the highest or lowest point of the parabola; the axis of symmetry, which is a vertical line that splits the parabola into two mirror images; and the roots or zeros of the function, which are the \( x \) values where the function intersects the x-axis.

In the exercise at hand, the quadratic function given is \( f(x) = (x-3)^2 \), which can be expanded to \( f(x) = x^2 - 6x + 9 \) if necessary. This is a specific case where the quadratic has been set to equal a perfect square binomial. Students examining this function should note that the vertex of this parabola is at \( (3, 0) \) and that it opens upwards since the coefficient of \( x^2 \) is positive. Understanding the characteristics of quadratic functions helps students predict their graphs and solve related problems more effectively.

Whether we are factoring, completing the square, or using the quadratic formula to find the roots, the study of quadratic functions is a cornerstone of algebra that extends into calculus and beyond, offering tools to model and solve real-world problems involving areas, projectiles, and many other applications.