The modulus function

Sketch the graph of y = | 2 x - 5 | on the grid provided in the Printed Answer Booklet.

Solve the inequality | 2 x - 5 | < 1.

In this question you must show detailed reasoning.

Hence find all the possible integer values N that satisfy the inequality

The function f (x) is defined for all values of x as f (x) = | ax - b |, where a and b are positive constants.

The graph of y = f (x) + c, where c is a constant, has a vertex at (3, 1) and crosses the y-axis at (0, 7).

Find the values of a, b and c.

Explain why f^-1(x) does not exist.

The function g(x) is defined for x ⩾ as g(x) = | px - q|, where p and q are positive constants.

Find, in terms of p and q, an expression for g^-1(x), stating the domain of g^-1(x).

State the set of values of p for which the equation g(x) = g^-1(x) has no solutions.

Solve the equation | 2 x - 3 | = 9.

Hence explain why the equation x|x - 1 | = k has exactly one real root for any negative value

of k.

Determine the real root of the equation x|x - 1 | =- 6.

In this question you must show detailed reasoning.

Solve the inequality | x - 2 | ⩽ | 2 x - 6 |.

Give full details of a sequence of two transformations needed to transform the graph of

y = | x - 2 | to the graph of y = | 2 x - 6 |.

Solve the equation | 2 x - 1 | = | x + 3 |.