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Showing Page:

1/3

The following function state the domain and range find all extreme values and the

function is increasing and decreasing and state where it is concave up and down and

locate any points of inflection and asymptotes. Then sketch a graph of the function noting

special points.

Consider the function

( )

2

9

x

f x

x

−

=

+

The denominator of the function is never zero, because

( )

2

, 9 0x x∈ + >¡

Therefore, the domain of

( )

2

9

x

f x

x

−

=

+

is

¡

To find range:

( )

2

9

x

f x

x

y

−

=

+

=

2

2

2

9

9 0

1 1 36

2

x

y

x

yx x y

y

x

y

−

=

+

+ + =

− ± −

=

Since

( )

2

, 1 36 0x y∈ − ≥¡

That implies

1 1

6 6

y− ≤ ≤

The range of

( )

2

9

x

f x

x

−

=

+

is

1 1

,

6 6

−

Since

( )

2

9

x

f x

x

−

=

+

Differentiating w.r.t

x

( )

( )

2

2

2

9

9

x

f x

x

−

′

=

+

The function

( )

2

9

x

f x

x

−

=

+

has extreme values if

( )

0f x

′

=

So,

( )

2

2

2

9

0

9

x

x

−

=

+

2

9 0

3

x

x

− =

= ±

And

( )

0f x

′

>

when

( )

2

9 0,x − >

because

( )

2

2

9 0x + >

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