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Question

  1. Let \(X = \{ ax + b : a,b\in\mathbb{R}\}\). Consider the function \(f:X\rightarrow \mathbb{R}\) given by \[ f~\left( ax + b\right) = (a+b)^2,\] for all \(a,b \in \mathbb{R}\). What is \(f(-2x + 1)\)?

  2. Consider the function \(f:\mathbb{R}^3\rightarrow \mathbb{R}^2\) given by \[ f~\left( \begin{bmatrix} x \\ y \\ z \end{bmatrix}\right) = \begin{bmatrix} 2x + y - z\\ x-y + 3z \end{bmatrix}.\] Let \(t = f~\left( \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}\right)\). What is the value of \(t_2\)?