91Ó°ÊÓ

When we talk about the "image of a set" under a transformation, we refer to the new collection of points in the coordinate space that result from applying the transformation rules to every point in the original set. In this context, the original set is composed of ordered pairs \(u, v\) bounded by certain conditions, such as \(0 \leqslant u \leqslant 3\) and \(0 \leqslant v \leqslant 2\).

In the exercise, the transformation is defined by the equations \(x = 2u + 3v\) and \(y = u - v\). Applying these, we take each permissible \(u, v\) pair and transform it into an \(x, y\) pair, thereby mapping the original set \(S\) into a new set in the \(xy\)-plane.

• To find the image, consider the boundary values of \(u\) and \(v\) first. These edges help define the limits for transformations and ensure all possibilities are explored.
• In this step-by-step process, it helps you visualize how each point shifts within the new coordinate frame.

By calculating and substituting all combinations that fit within the given constraints, the complete image shows how the set morphs into and spreads over a different region in the new space.

The range of a function pertains to all possible output values it can produce. For our transformation, this involves finding all possible \(x\) and \(y\) values that result from the defined transformations.

Starting with \(x = 2u + 3v\), determine the smallest and largest values by evaluating the boundaries of \(u\) and \(v\). The equations highlight that the lowest \(x\) value emerges when both variables are at their minimum (0), while the maximum occurs when both reach their upper limits. This provides the range \( 0 \leqslant x \leqslant 12\).
The process repeats for \(y = u - v\). Evaluating \( y\), we find the most negative possible value is when \(v\)'s value exceeds \(u\) substantially, and the peak when \(u\) is at its maximum and \(v\) is at its lowest. Thus, \( y \) spans from \(-2\) to \(3\).
This analysis of each function’s range allows mapping every possibility the transformation offers. Complete coverage justifies that we have accurately charted out the space that the transformed set occupies.

Coordinate transformations involve changing the original perspective of a set of points. They are crucial for representing scenarios in different spaces while preserving the relationships between points.

In the exercise, the transformation \((u, v)\) to \((x, y)\) means modifying each pair of input variables into a new, possibly more useful, form using specific equations: \(x = 2u + 3v\) and \(y = u - v\).

• The primary role of such transformations is to change the coordinate system for easier analysis or presentation of data.
• Analyzing the resulting image shows how transformations offer insights into the geometric shape and extent of an original set in its new format.

Each point in the initial set now corresponds uniquely to another point in these altered coordinates, retaining the structural integrity of relationships but projecting into a space that can reveal underlying properties or fit into differently scaled or aligned analyses.