Everything Maths Grade 10

Discovering the characteristics EMA4S

The standard form of a hyperbola is the equationy=

a

x

+q.

Domain and range

Fory=

a

x

+q, the function is undefined forx= 0. The domain is thereforefx:x 2 R; x̸= 0g.

We see thaty=

a

x

+qcan be rewritten as:

y=

a

x

+q

yq=

a

x

Ifx̸= 0then:(yq)x=a

x=

a

yq

This shows that the function is undefined only aty=q.

Therefore the range isff(x) :f(x) 2 R; f(x)̸=qg

Worked example 9: Domain and range of a hyperbola

QUESTION

Ifg(x) =

2

x

+ 2, determine the domain and range of the function.

SOLUTION

Step 1: Determine the domain

The domain isfx:x 2 R; x̸= 0gbecauseg(x)is undefined only atx= 0.

Step 2: Determine the range

We see thatg(x)is undefined only aty= 2. Therefore the range isfg(x) :g(x) 2 R; g(x)̸= 2g

Intercepts

They-intercept:

Every point on they-axis has anx-coordinate of 0, therefore to calculate they-intercept letx= 0.

For example, they-intercept ofg(x) =

2

x

+ 2is given by settingx= 0:

y=

2

x

+ 2

y=

2

0

+ 2

which is undefined, therefore there is noy-intercept.

Chapter 6. Functions 171