Use and apply the notation |𝑥| for the absolute value for the real number 𝑥 and the graph of 𝑦 = |𝑥|.

A function is defined piecewise as \( f(x) = |2x - 3| + |x + 1| \). The table below shows selected values of \( x \) and the corresponding values of \( f(x) \). Refer to the table below. (a) Determine the value of \( p \). (1 mark) (b) Determine the value of \( q \). (1 mark) (c) State the minimum value of \( f(x) \). (1 mark) (d) Determine the value of \( x \) for which \( f(x) \) attains its minimum value. (1 mark)

Consider the function \( f(x) = |2x - 6| \). a) Determine the value of \( x \) for which \( f(x) = 0 \). (1 mark) b) Calculate the value of \( f(-3) \). (1 mark) c) Use the definition of absolute value to express \( f(x) \) as a piecewise function in the form \[ f(x) = \begin{cases} \ldots & \text{if } x \geq \ldots \\ \ldots & \text{if } x < \ldots \end{cases} \] (1 mark)

What is the solution set of \(|2x - 4| = 6\)?

a) Use the definition of absolute value to express \(|2x - 6|\) as a piecewise function. (1 mark) b) Use your result from part (a) to determine the coordinates of the vertex of the graph \(y = |2x - 6|\). (1 mark) c) Use your results from parts (a) and (b) to state the value of \(\frac{dy}{dx}\) when \(x = 5\). (1 mark)

Use the graph of \(y = |x - 3| + 1\) shown below to determine the minimum value of the function.