Cambridge Additional Mathematics

13 x

y

O

A

B

1 x

y

O

y =ln(x ) 2

y=lnx

y

x

(1 3),

(3 1),

O

y=x

y=ex-3

y=3+lnx

Logarithms (Chapter 5) 151

Example 27 Self Tutor

Consider the function f:x 7 !ex¡^3.

a Find the equation defining f¡^1.

b Sketch the graphs of f and f¡^1 on the same set of axes.

c State the domain and range of f and f¡^1.

d Find any asymptotes and intercepts of f and f¡^1.

a f(x)=ex¡^3 b

) f¡^1 is x=ey¡^3

) y¡3=lnx

) y=3+lnx

So, f¡^1 (x)=3+lnx

c Function f f¡^1

Domain x 2 R x> 0

Range y> 0 y 2 R

d For f, the horizontal asymptote is y=0,

and they-intercept ise¡^3.

For f¡^1 , the vertical asymptote is x=0,

and thex-intercept ise¡^3.

2 For the following functionsf:

i Find the equation of f¡^1.

ii Sketch the graphs of f and f¡^1 on the same set of axes.

iii State the domain and range of f and f¡^1.

iv Find any asymptotes and intercepts of f and f¡^1.

a f(x)=ex+5 b f(x)=ex+1¡ 3

c f(x)=lnx¡ 4 , x> 0 d f(x) = ln(x¡1) + 2, x> 1

3 Consider the graphs A and B. One of them is the graph of

y=lnx and the other is the graph of y=ln(x¡2).

a Identify which is which. Give evidence for your answer.

b Copy the graphs onto a new set of axes and add to them

the graph of y=ln(x+2).

c Find the equation of the vertical asymptote for each graph.

4 Kelly said that in order to graph y=ln(x^2 ),x> 0 , you

could first graph y=lnx and then double the distance

of each point on the graph from thex-axis.

Is Kelly correct? Explain your answer.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_05\151CamAdd_05.cdr Friday, 20 December 2013 1:05:49 PM BRIAN