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]}