1.02l Modulus function: notation, relations, equations and inequalities

4. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.

1. Showing the coordinates of any points where the graph meets the axes, sketch the graph of \(y = | \mathrm { f } ( x ) |\). The function \(g\) is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
2. Find \(\mathrm { fg } ( \mathrm { a } )\) in terms of \(a\).
3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \rightarrow | 2 x - 5 | , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \ln ( x + 3 ) , \quad x \in \mathbb { R } , \quad x > - 3 \end{aligned}$$

1. State the range of f .
2. Evaluate fg(-2).
3. Solve the equation $$\operatorname { fg } ( x ) = 3$$ giving your answers in exact form.
4. Show that the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$ has a root, \(\alpha\), in the interval [3,4].
5. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left[ 5 + \ln \left( x _ { n } + 3 \right) \right]$$ with \(x _ { 0 } = 3\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
6. Show that your answer for \(x _ { 4 }\) is the value of \(\alpha\) correct to 4 significant figures.

4 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_529_737_900_701} The function f is defined by \(\mathrm { f } ( x ) = 2 - \sqrt { x }\) for \(x \geqslant 0\). The graph of \(y = \mathrm { f } ( x )\) is shown above.

1. State the range of f.
2. Find the value of \(\mathrm { ff } ( 4 )\).
3. Given that the equation \(| \mathrm { f } ( x ) | = k\) has two distinct roots, determine the possible values of the constant \(k\).

7 The curve \(y = \ln x\) is transformed to the curve \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.

1. Give full details of the translation and stretch involved.
2. Sketch the graph of \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\).
3. Sketch, on another diagram, the graph of \(y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|\).
4. State, in terms of \(a\), the set of values of \(x\) for which \(\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)\).

6 \includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_641_837_1306_657} The diagram shows the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).

1. Give details of the pair of geometrical transformations which transforms the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\) to the graph of \(y = \sin ^ { - 1 } x\).
2. Sketch the graph of \(y = \left| - \sin ^ { - 1 } ( x - 1 ) \right|\).
3. Find the exact solutions of the equation \(\left| - \sin ^ { - 1 } ( x - 1 ) \right| = \frac { 1 } { 3 } \pi\).

2 Find the exact solutions of the equation \(| 6 x - 1 | = | x - 1 |\).

2 Solve the inequality \(| 2 x - 3 | < | x + 1 |\).

2 Solve the inequality \(| 4 x - 3 | < | 2 x + 1 |\).

1 Find the exact solutions of the equation \(| 4 x - 5 | = | 3 x - 5 |\).

1 Solve the inequality \(| 2 x + 1 | > | x - 1 |\).

1 Fig. 1 shows the graphs of \(y = | x |\) and \(y = | x - 2 | + 1\). The point P is the minimum point of \(y = | x - 2 | + 1\), and Q is the point of intersection of the two graphs. \begin{figure}[h] \end{figure}

1. Write down the coordinates of P .
2. Verify that the \(y\)-coordinate of Q is \(1 \frac { 1 } { 2 }\).

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

2 Given that \(\mathrm { f } ( x ) = 1 - x\) and \(\mathrm { g } ( x ) = | x |\), write down the composite function \(\mathrm { gf } ( x )\).
On separate diagrams, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { gf } ( x )\).

1 Solve the inequality \(| 2 x - 1 | \leqslant 3\).

2 Given that \(\mathrm { f } ( x ) = | x |\) and \(\mathrm { g } ( x ) = x + 1\), sketch the graphs of the composite functions \(y = \mathrm { fg } ( x )\) and \(y = \operatorname { gf } ( x )\), indicating clearly which is which.

3 The graph shows part of the function \(y = a \ln ( b x )\). \includegraphics[max width=\textwidth, alt={}, center]{2f403099-2813-40d8-a9ae-1f7e64d41f80-2_377_762_900_685} The graph passes through the points \(( 2,0 )\) and \(( 4,1 )\).

1. Show that \(b = \frac { 1 } { 2 }\) and find the exact value of \(a\).
2. Solve the inequality \(| a \ln ( b x ) | < 2\).

2

1. Sketch the graph of \(y = | 2 x - 3 |\).
2. Hence, or otherwise, solve the inequality \(| 2 x - 3 | < 5\). Illustrate your answer on your graph.

3 Given that \(\mathrm { f } ( x ) = 1 - x\) and \(\mathrm { g } ( x ) = | x |\), write down the composite function \(\mathrm { gf } ( x )\). On separate diagrams, sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { gf } ( x )\).

1 Solve the equation \(| 3 - 2 x | = 4 | x |\).

2 Express \(1 < x < 3\) im th \(\quad | x - a | < b\), where \(a\) and \(b\) are to be determined.

3 Fig. 1 shows the graphs of \(y = | x |\) and \(y = a | x + b |\), where \(a\) and \(b\) are constants. The intercepts of \(y = a | x + b |\) with the \(x\)-and \(y\)-axes are \(( - 1,0 )\) and \(\left( 0 , \frac { 1 } { 2 } \right)\) respectively. \begin{figure}[h] \end{figure}

1. Find \(a\) and \(b\).
2. Find the coordinates of the two points of intersection of the graphs.

4 Solve the inequality \(| 2 x + 1 | \geqslant 4\).

5 Solve the equation \(| 2 x - 1 | = | x |\).
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7 Solve the inequality \(| x - 1 | < 3\).

8 Fig. 4 shows a sketch of the graph of \(y = 2 | x - 1 |\). It meets the \(x\) - and \(y\)-axes at ( \(a , 0\) ) and ( \(0 , b\) ) respectively. \begin{figure}[h] \end{figure} Find the values of \(a\) and \(b\).