- a.
- Suppose the quadratic function is given by . Are there any values that are never achieved as an output?
- b.
- Explain the difference in finding the domain of a function and finding the range of the function, if you are given the graph of the function. What if you’re given a formula for the function instead?
We examine the outputs that are actually achieved.
Motivating Questions
- If is a function from a set to a set , does every item in actually get related to something from ?
Introduction
In the last section, for a function from to , we called the set the domain and we called the set the codomain.
Let be the function defined by . We can consider this a function from the set of all real numbers to the set of all real numbers . In this case, the domain is and the codomain is also . We know that for any real number , the value of is never negative. That means there is no input to that ever gives a negative output.
Let be the function from the set of capital letters to the set of natural numbers, which assigns each letter to its placement in the alphabet. This means since ‘A’ is the first letter of the alphabet. Similarly and . In this case the domain is the set of capital letters and the codomain is the set of natural numbers . For the function there are only 26 capital letters in the alphabet, so no number past greater than 26 is ever an output of .
For both the function and just given, not every number in the codomain is actually achieved as the output of the function. There is a difference between the codomain, which measures the “possible outputs” and the actual outputs that are achieved.
The Range of a Function
This means the range consists of the outputs that are actually achieved. Not everything that is “possible”, but only those outputs that actually come out of the function. For each in the range of the function , there is actually an in the domain with .Let be the function defined by the following table. Find the range of .
The only outputs of this function are , , , and . (Even though the output is identified twice, it only gets counted once here.) There are no intervals, just the separated points. The range of is . Including individual numbers in curly brackets means we are considering the set whose only members are those individual numbers.
Let be the function given by . Is in the range of ? What about or ?
If we want to see if is in the range of , that means we need to check whether there is a number in the domain of with .
That means . We found an input that gives the output value , which means is in the range of .Let’s try the same calculation for .
That means , so that is in the range of .Now for :
This statement is false for all choices of . That means there is no value of with . In particular, is not in the range of .The entire graph of a function is given in the graph below. Find the range of .
Notice that as changes between to , the graph takes all outputs from to (not including the endpoints). As changes from to , all the numbers from to show up as outputs (including ). Together, this means that every number in the interval is in the range. The last two pieces of the graph have outputs and , which are already included in this interval.
The range is .
Let be the function given by . Find the range of .
We know the domain of is , so any real number is a valid input. We also know that
is a linear polynomial function (a polynomial of degree 1) so we know what its graph
is a straight line with slope . 
Since this graph goes higher and higher as travels to the right, and lower as travels
to the left, we believe that every number is eventually an output of this
function.
Let’s make a calculation to be sure that’s true by taking an arbitrary real number and show that it’s actually an output of . If and are real numbers with , then
That means that if is any arbitrary real number, then so that is achieved as an output of the function, so is in the range. In other words, this gives us a formula to identify an input that gives as an output. This means the range of is .
The Range of Famous Functions
- (a)
- The Absolute Value function - The average value of a number is never negative. The Absolute Value function has range .
- (b)
- Polynomial functions - This depends on the degree of the polynomial.
- (i)
- Odd degree - The range is .
- (ii)
- Even degree - We can only be precise with monomials (polynomials with
only one term) like or .
- i.
- If the monomial has positive coefficient, the range is .
- ii.
- If the monomial has negative coefficient, the range is .
- iii.
- If the monomial is a constant , the range is .
- (c)
- The Square Root function - Even-index radicals never have negative outputs. Their range is .
- (d)
- Exponential functions - Exponential functions , for with , have range . Notice that is never an output for these kinds of functions.
- (e)
- Logarithms - Logarithms have range .
- (f)
- The Sine function - The sine function has range .
For polynomials, if we know the turning points or maximum or minimum values
of the polynomial (as in the case of a parabola), you can determine the
range of the polynomial. Unfortunately, finding these in general requires
calculus.
Spotting Values not in the Range
Finding the range of a function is quite a bit more involved than finding the domain. Here are some guidelines if you are given a formula for the function instead of its graph.
- (a)
- The output of an even-index radical is never negative.
- (b)
- The output of is never negative if is an even natural number.
- (c)
- The output of an exponential function is always positive.
Find the range of the following functions
- (a)
- .
- (b)
- .
- (c)
- .
You’ll notice in these calculations, that finding the range of a function is a great deal more complicated than finding the domain, unless we have an accurate graph of the function, as above.
For each of these calculations, we will follow the same two steps. The idea is that the first step shows that the range is “no more than” the interval we build, and the second step shows that the range is “no less than” that interval. The only possibility left to us is that the range is exactly the interval we have constructed.
More specifically, in the first step will find a bound on the range. We’ll determine if any numbers are too big or too small to be a valid output of the function. This will give us an interval that the range will have to be inside. In the second step, we’ll see that everything inside that interval is actually attained by the function, by constructing an input value that gets assigned to that output.
- (a)
- Let’s start by looking at the graph of Notice that the graph of looks similar to what we know for , but upside down, stretched, and moved up. From the graph, we would say that the range should be . Let’s see how we can verify that with a few calculations.
We know that the range of is so for any in the domain of ,
The outputs of are never larger than , so the only numbers in the range are less than or equal to . That is, the range must be inside the interval .Metacognitive Moment Notice that this process is just looking at how function transformations are changing the range by undoing the transformations one at a time.To verify that the range is exactly , suppose , then:
For this value of , we have . That means is in the range of . That means every number in the interval is in the range of .
The range of is .
- (b)
- The graph of looks like this. This graph looks exactly like the graph of our famous function , just moved vertically upward by . The range of is , so our range should be shifted by to be .
To verify this with a calculation, the fact that has range means for any value of ,
The outputs of are greater than , so the range of is in the interval .To verify that the range is exactly , suppose , then:
Since the range of is , and , there is a value for with .(For instance, if , that would mean , so we would be looking to see if there is a value of with . This is illustrated in the graph below.)
For this value of , we have , meaning that is in the range of . The range of is .
- (c)
- The graph of is given below, from which it appears the range of is
.
The range of our famous function is . That means for any
The outputs of are in the interval . To verify that the range is exactly , suppose is a number with . Then: Because , we know . That means there is a number with , since the range of sine is .(For instance, if then so we would be looking for a value of with . We know that , so this means we’d take .) For this value of , we have , so is in the range of . The range of is then .