# How do you determine the intervals for which the function is increasing or decreasing given #f(x)=(x^2+5)/(x-2)#?

##### 2 Answers

See the explanation.

#### Explanation:

By actual division,

f(x) = y = x+2+9/(x-2)

For

For

For

For x

The complexity in rise and fall of y is understandable upon seeing

that the given equation has the form

This represents the hyperbola having asymptotes

( slant ) y = x +2 and ( vertical ) x = 2.

Respectively, there is rise and fall in the two branches.

See the illustrative graph.

graph{(y-x-2)(x-2)-9=0 [-80, 80, -40, 40]}

In my style, this is my answer. There ought to be some omissions or

additions, and corrections there upon, for improvement.

I request ( 1 other ) editors to give all that in comments, separately. It

is my duty to thank them, and edit my answer, accordingly..

The function is increasing when

The function is decreasing when

#### Explanation:

The domain of

We take the derivative of

The derivative of a polynomial fraction is

Here, we have

so,

Therefore,

The critical points are

The denominator is

Now we can form the sign chart

Therefore,

The function is increasing when

The function is decreasing when