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

11 The functions f and g are defined by \(\mathrm { f } ( x ) = \frac { 1 } { 2 + x } + 5 , x > - 2\) and \(\mathrm { g } ( x ) = | x | , x \in \mathbb { R }\).

1. Given that the range of f is of the form \(\mathrm { f } ( x ) > a\), find \(a\).
2. Find an expression for \(\mathrm { f } ^ { - 1 }\), stating its domain and range.
3. Show that \(\mathrm { gf } ( x ) = \mathrm { f } ( x )\).
4. Find an expression for \(\mathrm { fg } ( x )\). Determine whether fg has an inverse.

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

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

3 The function f is given by \(\mathrm { f } ( x ) = | x - 2 | + 3\) for \(- 5 \leqslant x \leqslant 5\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Explain why f is not one-one.

5 Solve \(| x - \sqrt { 3 } | < | x + 2 \sqrt { 3 } |\) giving the answer in exact form.

1 Solve \(5 x + 3 < | 3 x - 1 |\).

1 Find the set of all real values of \(x\) which satisfy the equation $$| 2 x + 5 | < 7$$

The \(n\)th term of a number sequence is denoted by \(x _ { n }\). The \(( n + 1 )\) th term is defined by \(x _ { n + 1 } = 4 x _ { n } - 3\) and \(x _ { 3 } = 113\). a) Find the values of \(x _ { 2 }\) and \(x _ { 1 }\).
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.
a) Express \(5 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find the minimum value of \(\frac { 4 } { 5 \sin x - 12 \cos x + 15 }\).
c) Solve the equation $$5 \sin x - 12 \cos x + 3 = 0$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). a) Find the range of values of \(x\) for which \(| 1 - 3 x | > 7\).
b) Sketch the graph of \(y = | 1 - 3 x | - 7\). Clearly label the minimum point and the points where the graph crosses the \(x\)-axis.

Solve the inequality \(|5x + 7| > |2x - 3|\). [4]

Solve the inequality \(|x - 7| > 4x + 3\). [4]

Find the set of values of \(x\) satisfying the inequality \(|8 - 3x| < 2\). [3]

Solve the inequality \(2x > |x - 1|\). [4]

Solve the inequality \(|x - 3| > 2|x + 1|\). [4]

Solve the equation \(|x - 2| = |\frac{1}{3}x|\). [3]

Find the set of values of \(x\) satisfying the inequality $$|x + 2a| > 3|x - a|,$$ where \(a\) is a positive constant. [4]

Solve the inequality \(|x - 3| < 3x - 4\). [4]

Solve the equation \(2|3^x - 1| = 3^x\), giving your answers correct to 3 significant figures. [4]

Solve the inequality \(3|2x - 1| > |x + 4|\). [4]

Find the set of values of \(x\) satisfying the inequality \(2|2x - a| < |x + 3a|\), where \(a\) is a positive constant. [4]

The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).

1. Find the equations of the asymptotes of \(C\). [2]
2. Find the coordinates of any stationary points on \(C\). [4]
3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]