Starter

The functions f and g are defined by

Find an expression for gf in terms of x and state its domain and range.

Sketch the graph of y = gf(x) showing the coordinates of any points of intersections with the coordinate axes.

Show that, if the equation gf(x) – 2f(x) = a has no real roots, then a < –10.

x,9xx:g

x,1)2(xx:f2

Objective:

to form inverse functions

xef(x)

xln(x)f -1

Inverse Functions

A function f cannot have an inverse function f -1 unless f is one-to-one (there must be a unique value y in the range of f for each value of x in the domain of f).

The range of f is the domain of f -1 and vice versa.

xef(x)

xln(x)f -1

Inverse Functions

The graphs of a pair of inverse functions are reflections of each other in the line y = x.

Examples

Find the inverse of the functions f and g.

2x,x,2x2x

x:g

x,52xx:f

1x,x,1x1)2(x

x:g

x,2

5xx:f

1-

1-

Core 3 & 4 Textbook

Exercise 2C

Page 35

Questions12 to 18