The function f is defined by \(\mathrm{f}(x)=2 x-1\) for \(x \in \mathbb{R}\). (a) Explain why the function \(\mathrm{f}^{2}\) can be formed.

(b) On the axes, sketch the graph of \(y=\left|\mathrm{f}^{2}(x)\right|\).

State any intercepts with the coordinate axes.

(c) It is given that \(\left|\mathrm{f}^{2}(x)\right| \leqslant a x+b\) for \(-1 \leqslant x \leqslant 3\) and for no other values of \(x\).

Find the values of \(a\) and \(b\).

(a) On the axes, sketch the graph of \(y=|4 x-6|\), showing the points where the graph meets the axes.

(b) Solve the equation \(|4 x-6|=|2 x|\).

The diagram shows the graphs of \(y=|f(x)|\) and \(y=g(x)\), where \(y=f(x)\) and \(y=g(x)\) are straight lines. Solve the inequality \(|f(x)|\le g(x)\).

(a) On the axes, sketch the graphs of

\(y=\left|2x+1\right|\quad\text{and}\quad y=\left|5-3x\right|\)

for \(-2\leq x\leq 8\). State the coordinates of the points where these graphs meet the coordinate axes.

(b) Solve the equation

\(\left|2x+1\right|=\left|5-3x\right|.\)

(a) On the axes, draw the graphs of

\(y=5+|3x-2|\quad\text{and}\quad y=11-x.\)

(b) Using the graphs, or otherwise, solve the inequality

\(11-x\lt 5+|3x-2|.\)

(a) On the axes draw the graphs of

\(y=|x-5|\quad\text{and}\quad y=6-|2x-7|.\)

(b) Use your graphs to solve the inequality

\(|x-5|\gt 6-|2x-7|.\)

The function \(f\) is defined by

\(f(x)=|5x-7|.\)

(a) Sketch the graph of \(y=f(x)\), showing the coordinates of the points where the graph meets the axes.

(b) Solve the equation

\(5|5x-7|-1=14.\)

(i) Sketch the graph of \(y=|5x-3|\), showing the coordinates of the points where the graph meets the coordinate axes.

(ii) Solve the equation \(|5x-3|=2-x\).

(i) Draw the graph of

\(y=|2x-3|.\)

(ii) Solve the equation

\(7-|2x-3|=0.\)

(i) Sketch the graph of \(y=\left|6-3x\right|\), showing the coordinates of the points where the graph meets the coordinate axes.

(ii) Solve \(\left|6-3x\right|=2\).

(iii) Hence find the values of \(x\) for which \(\left|6-3x\right|\gt 2\).

(i) On the axes, sketch the graphs of \(y=|2x-5|\) and \(9y=80x-16x^2\).

(ii) Solve \(|2x-5|=4\).

(iii) Hence show that the graphs of \(y=|2x-5|\) and \(9y=80x-16x^2\) intersect at the points where \(y=4\).

(iv) Hence find the values of \(x\) for which \(9|2x-5|\le 80x-16x^2\).