91Ó°ÊÓ

The image of a function refers to the set of all possible output values it can produce. In mathematical terms, if you have a function like \( f(x, y) \), its image is the collection of all points that can be reached by the function applied to all permissible inputs. For the given function \( f(x, y) = \begin{pmatrix} \sin x \cosh y \ \cos x \sinh y \end{pmatrix} \), calculating the image involves determining what outputs are possible when \( (x, y) \) varies over specified domains like \( G_1 \) and \( G_2 \).

• For \( G_1 \), the image is in the first quadrant, since both function components are positive for the values between zero and \( \frac{\pi}{2} \).
• For \( G_2 \), the scenario shifts to the fourth quadrant due to the sign change in \( \cos x \) when \( x \) is between \( \frac{\pi}{2} \) and \( \pi \).

Understanding the image is crucial when analyzing whether a function, like \( f \), is injective under certain conditions.

Trigonometric functions are mathematical functions relating to angles and ratios found in right triangles. The main ones are sine, cosine, tangent, and their reciprocals. The function \( f(x, y) \) employs two of these: sine and cosine. These functions exhibit periodic behavior, alternating between positive and negative values as the angle changes.

• The sine function, \( \sin x \), reaches its peak value at \( \frac{\pi}{2} \) and is positive as \( x \) increases from 0 to \( \pi \).
• The cosine function, \( \cos x \), decreases from 1 to 0 in the interval \( 0 \) to \( \frac{\pi}{2} \) and becomes negative beyond \( \frac{\pi}{2} \).

These properties are essential when evaluating how the function's image is positioned in different quadrants, as seen in the original exercise.

Much like trigonometric functions but related to hyperbolas, hyperbolic functions include \( \sinh \) (hyperbolic sine) and \( \cosh \) (hyperbolic cosine). These functions appear in the given function \( f(x, y) \). They are defined as:

• \( \sinh y = \frac{e^{y} - e^{-y}}{2} \): similar to the sine wave but not periodic.
• \( \cosh y = \frac{e^{y} + e^{-y}}{2} \): always positive, contributing to the function's range.

Hyperbolic functions are critical when describing the growth behavior of the function as \( y \) changes, especially because both \( \sinh y \) and \( \cosh y \) affect the image by always remaining positive when \( y \) is positive, which helps in maintaining injectivity for separate regions \( G_1 \) and \( G_2 \).

Plane geometry involves shapes and figures residing in a two-dimensional space, like the strips described as \( G_1 \) and \( G_2 \). In this exercise, \( G_1 \) and \( G_2 \) are sections of the Cartesian plane defined by specific ranges of \( x \) values. Focusing on these sections involves understanding how these planes get transformed under \( f(x, y) \).

• \( G_1 \): a horizontal band with values \( 0 < x < \frac{\pi}{2} \).
• \( G_2 \): another band but shifted to \( \frac{\pi}{2} < x < \pi \).

Both strips represent a critical aspect of the injectivity and overall mapping behavior of \( f \). By constructing how outputs differ across these strips, we uncover why \( f \) fails to be injective over their union, yet remains injective on each separately. This visualization is an important part of understanding plane geometry's role in function mappings.