Answer true or false to each of the following statement and explain your answer.

Give a counterexample to show that the given transformation is not a linear transformation.
$T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}xy\\ x+y\end{array}\right]T$

Provide answers to all tasks using the information provided.
a) Find the parent function f.
Given Information: $g\left(x\right)=-2|x-1|-4$
b) Find the sequence of transformation from f to g.
Given information: $f\left(x\right)=\left[x\right]$
c) To sketch the graph of g.
Given information: $g\left(x\right)=-2|x-1|-4$
d) To write g in terms of f.
Given information: $g\left(x\right)=-2|x-1|-4\text{and}f\left(x\right)=\left[x\right]$

What do you mean by the Karl Pearsons’ coefficient of correlation? What are the properties of the coefficient of correlation?

To find: The solution of the given initial value problem.

Answer true or false to each of the following statement and explain your answer.
It is not always possible to fnd a power transformation of the response variable or the predictor variable (or both) that will straighten the scatterplot.

To determine the solution of the initial value problem
$y{}^{″}+4y=\sin t+u_{\pi }\left(t\right)\sin (t-\pi ):$
$y(0)=0,$
$y^{\prime }(0)=0.$
Also, draw the graphs of the solution and of the forcing function and explain the relation between the solution and the forcing function..

Explain how to find the value of ${\displaystyle \cos (-{45}^{\circ })}$ using even-odd properties.