Question

Let $$\mathbf{F}(x, y)=\left[\begin{array}{c} x^{2}-y^{2} \\ 2 x y \end{array}\right]$$ (Example 6.3 .4 ) and $$S=\\{(x, y) \mid a x+b y>0\\} \quad\left(a^{2}+b^{2} \neq 0\right)$$ Find \(\mathbf{F}(S)\) and \(\mathbf{F}_{S}^{-1} .\) If $$S_{1}=\\{(x, y) \mid a x+b y<0\\}$$ show that \(\mathbf{F}\left(S_{1}\right)=\mathbf{F}(S)\) and \(\mathbf{F}_{S_{1}}^{-1}=-\mathbf{F}_{S}^{-1}\).

Short answer

To summarize, we began with the function $\mathbf{F}(x, y)$ and regions $S$ and $S_1$. We found that $\mathbf{F}(S_1)$ is equal to $\mathbf{F}(S)$, and that the inverse functions restricted to the two regions are negatives of each other. The primary reason for obtaining this result is the symmetry of both the function $\mathbf{F}(x, y)$ and the regions $S$ and $S_1$ with respect to the origin.

Step-by-step solution

Familiarize with the given function and region

We are given the function $$\mathbf{F}(x, y)=\left[\begin{array}{c} x^{2}-y^{2} \\\ 2 x y \end{array}\right]$$ and region \(S=\){(x, y) | ax+by>0} where \(a^2 + b^2 \neq 0\).

Find \(\mathbf{F}(S)\)

To find the image of the region S under the transformation F, we substitute x and y in the function \(F(x, y)\): \(\mathbf{F}(S) = \{ \mathbf{F}(x, y) | ax + by > 0 \}\)

Invert the function \(\mathbf{F}(x,y)\)

In order to compute the inverse function \(\mathbf{F}_{S}^{-1}\), we need first to invert the function \(\mathbf{F}(x, y)\): Let $$\mathbf{F}^{-1}(u, v)=\left[\begin{array}{c} x \\\ y \end{array}\right]$$ We need to find x and y in terms of u and v by solving the system of equations: $$u = x^2 - y^2$$ $$v = 2xy$$ Divide the second equation by 2: $$\frac{v}{2} = xy$$ Square both equations and add them: $$u^2 + \frac{v^2}{4} = (x^2 - y^2)^2 + (2xy)^2$$ $$u^2 + \frac{v^2}{4} = x^4 + 2x^2y^2 + y^4$$ Now, substitute \(xy = \frac{v}{2}\): $$u^2 + \frac{v^2}{4} = x^4 + \frac{v^2}{2} x^2 + y^4$$ Rearrange and factor this equation: $$x^4 + \frac{v^2}{2}x^2 + y^4 - u^2 -\frac{v^2}{4} = 0$$ $$x^2\left(x^2 + \frac{v^2}{2}\right) + y^4 - u^2 - \frac{v^2}{4} = 0$$ From the first equation \(u = x^2 - y^2\), we can express x^2 or y^2 in terms of u: $$x^2 = u + y^2$$ Plug in the expression for x^2 into the factored equation: $$(u + y^2)\left(\frac{v^2}{2} + u + y^2\right) + y^4 - u^2 - \frac{v^2}{4} = 0$$ This is an implicit equation for the inverse function \(\mathbf{F}^{-1}(u,v) = (x,y)\).

Find \(\mathbf{F}_{S}^{-1}\)

To find \(\mathbf{F}_{S}^{-1}\), restrict the domain of the inverse function by requiring \(ax + by > 0\): $$\mathbf{F}_{S}^{-1} = \{ (u,v) | (u + y^2)\left(\frac{v^2}{2} + u + y^2\right) + y^4 - u^2 - \frac{v^2}{4} = 0 \land ax + by > 0\}$$

Compute \(\mathbf{F}(S_1)\) and \(\mathbf{F}_{S_1}^{-1}\)

For region \(S_1\), we are tasked to show that \(\mathbf{F}(S_1)\) is equal to \(\mathbf{F}(S)\) and that \(\mathbf{F}_{S_1}^{-1}\) is the negation of \(\mathbf{F}_{S}^{-1}\). By symmetry of the function and the regions, we can observe that since \(S_1\) is defined by the linear inequality \(ax + by < 0\), it is the reflection of the region \(S\) around the origin. Hence, the image of \(S_1\) under the transformation \(\mathbf{F}\) is the same as the image of \(S\), because the function \(\mathbf{F}(x, y)\) is symmetric with respect to the origin as well. Thus, \(\mathbf{F}(S_1) = \mathbf{F}(S)\). Now, as the inverse function is also symmetric with respect to the origin, the inverse restricted to \(S_1\) is the negation of the inverse restricted to \(S\). Thus, we arrive at the desired result: \(\mathbf{F}_{S_1}^{-1} = -\mathbf{F}_{S}^{-1}\).

Key concepts

Function Transformation

Understanding function transformation is essential when studying various mathematical concepts, particularly when analyzing how particular operations alter the output of a function. A function transformation takes a given function and modifies it into another function that has been shifted, scaled, reflected, or rotated in some way.

For instance, consider the function \( F(x, y) = \left[\begin{array}{c} x^{2}-y^{2} \ 2xy \end{array}\right] \). When we refer to \( F(S) \), we mean the set of all points that result from applying the function F to each point in the set S. To do this, we need to substitute each pair (x, y) in S into the function F, thus transforming the original region S into a new set of points.

Moreover, the problem illustrates how the region defined by a linear inequality is transformed. The inequality \( ax+by>0 \) forms the region S, and applying F to S would give us the transformed region, \( F(S) \). This region is consequently influenced by the nature of the function itself—whether it expands, contracts, or flips the original set of points in S.

Inverse Functions

An inverse function essentially reverses the effects of the original function. For a given bijective function \( f(x) \), the inverse function, denoted as \( f^{-1}(x) \), will yield the original x-value when applied to the function's output y-value. In simpler terms, if you plug the output of the function into its inverse, you'll get back the starting number.

In our example, finding the inverse function \( \mathbf{F}_{S}^{-1} \) required solving a system of equations derived from the original function \( \mathbf{F}(x, y) \) to express x and y in terms of new variables u and v. The inverse function's behavior gives us insight into the original function's characteristics and understanding this relationship is a crucial aspect of real analysis.

The existence and determination of an inverse function depend on the function being one-to-one. If the original function maps two different inputs to the same output, an inverse cannot be properly defined because there would be ambiguity as to which input should be returned.

Symmetry in Mathematical Functions

Symmetry in mathematical functions refers to the invariance of the function's form when the input or the coordinate system is altered in some way. This can include reflections across an axis or the origin, rotations, and other transformations that leave the function unchanged.

In our textbook example, symmetry comes into play when we're asked to show that \( \mathbf{F}(S_{1}) \) is the same as \( \mathbf{F}(S) \) and \( \mathbf{F}_{S_{1}}^{-1} \) is the negation of \( \mathbf{F}_{S}^{-1} \). The function \( \mathbf{F}(x, y) \) is symmetric with respect to the origin, meaning that it remains unchanged under rotation by 180 degrees or reflection across the origin. As a result, the image of the region reflected across the origin (\(S_1\)) under \( \mathbf{F} \) will be the same as that of the original region (S), despite the regions themselves being different.

Identifying symmetries in functions can simplify the analysis and make solving problems more intuitive, as it can reduce the amount of computation needed and help visualize the behavior of functions and sets.