Composite and inverse functions

The point P(6, –4) lies on the curve with equationy = f(x), x Find the point to which P is mapped when the curve with equationy = f(x) is transformed to the curve with equation

y = f(x + 2)

y = f^–1(x)

y = 2f(x) – 3

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.

The functions f and g are defined by

Find fg(2)

Find g^–1

Find gf(x), giving your answer as a simplified fraction.

Deduce the range of gf(x).

The function h is defined by

where k is a constant.

Find the range of values of k for which the equation

f(x) = h(x)

has no real solutions.

The function f is defined by

State the range of f

Find f^–1

The function g is defined by

Find gf(6)

Find the exact value of the constant a for which

The function f is defined by

Show that

where A and B are constants to be found. The function g is defined by

State the range of g⁻¹

Find the range of f g⁻¹

The functions f and g are defined by

State the range of f

Find gf (1.8)

Find g^–1(x)

The functions f and g are defined by

State the range of f

Find gf (1.8)

Find g^–1(x)

The function f is defined by

Find f⁻¹(7)

Show that ff (x) = where a and b are integers to be found.

Figure 4 shows a sketch of the graph of y[࣠), where

Find the value of gg(0).

Find all values of x for which

The function h is defined by

Explain why h has an inverse but g does not.

Solve the equation

Find gg(5).

State the range of g.

Find g¹(x), stating its domain.

Figure 2 shows a sketch of part of the curve with equation

Show that g' (x) = f(x)e^–2x, where f(x) is a cubic function to be found.

Hence find the range of g.

State a reason why the function g^–1(x) does not exist.

Find the range of g.

Find the exact value of a for which g(a) = g^–1(a).

The function f is defined by

State the range of f.

Find f^–1 and state its domain.

The function g is defined by

Solve the equation

giving your answer in its simplest form.

Find fg(x), giving your answer in its simplest form.

Find, in terms of the constant k, the solution of the equation

The functions f and g are defined by

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

Solve the equation

The function f has domain –2 x 6 and is linear from (–2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1.

Write down the range of f.

Find ff(0).

The function g is defined by

Solve the equation gf(x) = 16

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.

The functions f and g are defined by

State the range of f.

Find fg(x), giving your answer in its simplest form.

Find the exact value of x for which f (2x 3) 6

Find f ¹, the inverse function of f, stating its domain.

On the same axes sketch the curves with equation y = f (x) and y = f ¹(x), giving the coordinates of all the points where the curves cross the axes.

The function f is defined by

Find the domain of f ¹.

The function g is defined by

Find fg (x), giving your answer in its simplest form.

Find the range of fg.

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.

Figure 2 shows a sketch of the curve with the equation y = f (x), The curve has a turning point at A (3, 4) and also passes through the point (0, 5).

Write down the coordinates of the point to which A is transformed on the curve with equation

Sketch the curve with equation

On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the y-axis.

The curve with equation y f (x) is a translation of the curve with equation y = x².

Find f (x).

Explain why the function f does not have an inverse.

Find the exact solutions to the equations

ln (3 x – 7) = 5

The functions f and g are defined by

Find f^–1 and state its domain.

Find fg and state its range.

The curve meets the coordinate axes at the points A(0,1–k) and B (–¹₂ k ln,0), where k is a constant and k 1, as shown in Figure 2. On separate diagrams, sketch the curve with equation

y = f ¹(x).

Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.

Given that f (x) = e^2x − k,

state the range of f,

write down the domain of f ¹.

The functions f and g are defined by

Write down the range of g.

Show that the composite function fg is defined by

Write down the range of fg.