Functions

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drawing

The functions g and h are defined as follows:

Find the following functions, giving the domain and range of each.

(a) Sketch the graph of .
(b) Hence, or otherwise, solve each of the following equations:
(i)
(ii)
The diagram below shows the graph , where for .

The graph approaches the line

as

becomes very large.

(a) Write down the domain and range of

.
(b) Find the inverse function

.

Write down the domain and range of

.
(d) Sketch the graph of

for the domain you gave in (iii).
(e) What is the relationship between the graph of

and the graph of

A biologist observes a population of rabbits regularly over a period of several years. Her observations show that for the first 2 years the population size doubles approximately every six months, from an initial population of 50 rabbits, before levelling off. She suggests the following model for the population in the first 2 years:

where

is the number of rabbits,

is the time in years and

and

are constants.
(a) Give values for

and

, explaining your reasoning carefully.

The biologist is interested in how much food is required to sustain the rabbit population. She suggests a model of:

where

is the number of units of food consumed daily by the population,

is the number of rabbits in the population and

is a constant.
(b) Explain what the coefficient of

represents in this model.

State one assumption made by the biologist.
(d) Explain why the given domain for the function is unlikely to be a good model.
(e) Find the composite function

in terms of

, stating its domain clearly.

Explain what relationship is given by this composite function.
(f) Explain why it does not make sense to find the composite function

.
5. The graph of a function

is shown below. The graph has a local maximum at

and a local minimum at

.

Sketch each of the following graphs, giving the coordinates of the turning points in each case.
(a)

(b)

(d)

drawing