Composite and inverse functions

Functions f and g are defined for all real values of x by

You are given that the equation f^-1g(x) = 4 - 2kx has real distinct roots.

Show that k satisfies the inequality 2k² - k - 6 > 0.

In this question you must show detailed reasoning.

Hence find the set of values of k. Give your answer in set notation.

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.

Give full details of the single transformation that transforms the graph of y = x³ to the graph

The function f is defined by f(x) = x³ - 8.

Find an expression for f^-1(x).

State how the graphs of y = f(x) and y = f^-1(x) are related geometrically.

The graph of the function y = f (x) passes through the point P with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to P on each of the following curves.

y = f (x) + 3

y = 2f (3 x - 1)

y = f^-1(x)

The diagram shows part of the graph of y = g ′ (x). This is the graph of the gradient function of y = g (x). The graph intersects the x-axis at x =- 2 and x = 4.

State the x-coordinate of any stationary points on the graph of y = g (x).

State the set of values of x for which y = g (x) is a decreasing function.

State the x-coordinate of any points of inflection on the graph of y = g (x).

Functions f and g are defined for 0 ⩽ x ⩽ 2 π by f (x) = 2 tan x and g (x) = sec x.

State the range of f.

State the range of g.

Show that fg (0.6) = 5.33, correct to 3 significant figures.

Explain why f^-1g(0.6) is not defined.

The functions f and g are defined for all real values of x by

f(x) = 2x² + 6x and g(x) = 3x + 2.

Find the range of f.

Give a reason why f has no inverse.

Given that fg(-2) = g^-1(a), where a is a constant, determine the value of a.

Determine the set of values of x for which f (x) > g (x). Give your answer in set notation.

The function f is defined by f(x) = (x - 3)² - 17 for x ⩾ k, where k is a constant.

Given that f^-1(x) exists, state the least possible value of k.

Evaluate ff(5).

Solve the equation f (x) = x.

Explain why your solution to part (c) is also the solution to the equation f(x) = f^-1(x).

In this question you must show detailed reasoning.

The functions f and g are defined for all real values of x by

Write down expressions for

fg (x),

gf (x).

Hence find the values of x for which fg (x) - gf (x) = 24.