Cambridge Additional Mathematics

-2 2 4 6 8 10

6

4

2

-2

y

O x

x=1

(2 1),

(5 3),

y =logw(x - 1) + 1

1_Qw

150 Logarithms (Chapter 5)

Example 26 Self Tutor

Consider the function f(x) = log 2 (x¡1) + 1.

a Find the domain and range off.

b Find any asymptotes and axes intercepts.

c Sketch the graph offshowing all important features.

d Find f¡^1.

a x¡ 1 > 0 when x> 1

So, the domain is fx:x> 1 g and the range is y 2 R.

b As x! 1 from the right, y!¡1, so the vertical asymptote is x=1.

As x!1, y!1.

When x=0, yis undefined, so there is noy-intercept.

When y=0, log 2 (x¡1) =¡ 1

) x¡1=2¡^1

) x=1^12

So, thex-intercept is 112.

cdf is defined by y= log 2 (x¡1) + 1

) f¡^1 is defined by x= log 2 (y¡1) + 1

) x¡1 = log 2 (y¡1)

) y¡1=2x¡^1

) y=2x¡^1 +1

) f¡^1 (x)=2x¡^1 +1

which has the horizontal asymptote y=1X

Its domain is fx:x 2 Rg, and

its range is fy:y> 1 g.

EXERCISE 5H

1 For the following functionsf:

i Find the domain and range.

ii Find any asymptotes and axes intercepts.

iii Sketch the graph of y=f(x) showing all important features.

iv Solve f(x)=¡ 1 algebraically and check the solution on your graph.

v Find f¡^1.

a f:x 7 !log 3 (x+1), x>¡ 1 b f:x 7! 1 ¡log 3 (x+1), x>¡ 1

c f:x 7 !log 5 (x¡2)¡ 2 , x> 2 d f:x 7! 1 ¡log 5 (x¡2), x> 2

e f:x 7! 1 ¡2 log 2 x, x> 0

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_05\150CamAdd_05.cdr Friday, 31 January 2014 10:39:54 AM GR8GREG