# What is the domain and range of #f(x) = (x+3)/(x^2+4)#?

##### 3 Answers

Domain: the whole real line

Range:

#### Explanation:

This question can be interpreted in one of two ways. Either we expect to only deal with the real line

The domain of

The equation

To determine the range of

Take the first derivative via the quotient rule:

The function

We solve this by the quadratic formula:

We characterise these points by examining their values at the second derivative of

We know from our first derivative root calculation that the second term in the numerator is zero for these two points, as setting that to zero is the equation we just solved to find the input numbers.

So, noting that

In determining the sign of this expression, we ask whether

So the sign of the whole expression comes down to the

So now to obtain the range, we must calculate the values of the function at the minimum and maximum points

Recall that

So, over the real line

Plot the graph of the function as a sanity check:

graph{(x+3)/(x^2+4) [-15, 4.816, -0.2, 1]}

Domain:

Range:

#### Explanation:

Given

**Domain**

The **domain** are all values of

For any function expressed as a polynomial divided by a polynomial, the function is defined for all values of

**Range**

The **range** is a little more interesting to develop.

We note that if a continuous function has limits, the derivative of the function at the points resulting in those limits is equal to zero.

Although some of these steps may be trivial, we will work through this process from fairly basic principles for derivatives.

[1] **Exponent Rule for Derivatives**

If

[2] **Sum Rule for Derivatives**

If

[3] **Product Rule for Derivatives**

If

[4] **Chain Rule for Derivatives**

If

~~~~~~~~~~~~~~~~~~~~

For the given function

we note that this can be written as

By [3] we know

By [1] we have

and by [2]

By [4] we have

and by [1] and [2]

or, simplified:

giving us

which can be simplified as

As noted (way back) this means that the limit values will occur when

then using the quadratic formula (look this up, Socratic is already complaining about the length of this answer)

when

Rather than prolong the agony, we will simply plug these values into our calculator (or spreadsheet, which is how I do it) to get the limits:

and

A simpler way of finding the range. The domain is

#### Explanation:

The domain is

Let

Cross multiply

This is a quadratic equation in

There are solutions if the discriminant

Therefore,

The solutions of this inequality are

graph{(x+3)/(x^2+4) [-6.774, 3.09, -1.912, 3.016]}