The modulus function

Figure 3 shows a sketch of the graph with equation y = f (x) where f (x) = 4 x – 3 – 5 x Given that a is a constant and a = 1

find the possible values of f (a) The function g is defined by g (x) = 2x + 17 x

Find the range of g f (x) The function h is defined by h (x) = kx x where k is a constant. Given that the equation f (x) = h (x) has no solutions,

find the range of values of k.

Figure 4 shows a sketch of the graph with equation y = | 2 x – 3 k | where k is a positive constant.

Sketch the graph with equation y = f (x) where f (x) = k – | 2 x – 3 k | stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

Figure 4 shows a sketch of the graph with equation

y = | 2 x – 3 k |

where k is a positive constant.

Sketch the graph with equation y = f (x) where

f (x) = k – | 2 x – 3 k |

stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of k, the set of values of x for which

giving your answer in set notation.

Find, in terms of k, the coordinates of the minimum point of the graph with equation

Given that

sketch, on separate diagrams, the curve with equation

y = f(x)

y = |f(x)| On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote.

Deduce the set of values of x for which f(x) = |f(x)|

Find the exact solutions of the equation |f(x)| = 2

Figure 1 shows part of the graph with equation y = f (x), x . The graph consists of two line segments that meet at the point Q (6, –1). The graph crosses the y-axis at the point P (0, 11). Sketch, on separate diagrams, the graphs of

y = | f (x) |

y = 2f (– x) + 3

On each diagram, show the coordinates of the points corresponding to P and Q. Given that f (x) = a | x – b | – 1, where a and b are constants,

state the value of a and the value of b.

Sketch the graph with equation y = | 4 x – 3 | stating the coordinates of any points where the graph cuts or meets the axes.

Find the complete set of values of x for which

Given that f(x) = ln x, x > 0 sketch on separate axes the graphs of

y = f(x),

y = | f(x) |,

y = –f(x – 4). Show, on each diagram, the point where the graph meets or crosses the x-axis. In each case, state the equation of the asymptote.

The functions f and g are defined by

Find g", the inverse function of g. (2)

Solve the equation

Figure 1 shows part of the curve with equation y = f (x), x . The curve passes through the points Q(0,2) and P(3,0) as shown.

Find the value of ff (3).

On separate diagrams, sketch the curve with equation

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

Figure 2 shows part of the curve with equation y = f (x) The curve passes through the points P(1 5., 0) and Q(0, 5) as shown. On separate diagrams, sketch the curve with equation

y = 2f (3x)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

Figure 1 shows part of the graph of y = f (x), x . The graph consists of two line segments that meet at the point R (4, – 3), as shown in Figure 1. Sketch, on separate diagrams, the graphs of

y = 2f (x + 4),

y = f (x).

On each diagram, show the coordinates of the point corresponding to R.

The function f is defined by

Sketch the graph with equation y = f (x), showing the coordinates of the points where the graph cuts or meets the axes.

Solve f (x) =15 + x.

The function g is defined by

Find fg(2).

Find the range of g.