- a.
- Suppose the quadratic function is given by . Are there any values that can’t be plugged into ?
- b.
- Suppose a square has side length denotes by the variable , and area denoted by . The area of the square is a function of the side length, . Are there any values of that don’t make sense?
- c.
- Suppose that is the rational function given by and that is the constant function given by . Are these the same function? Why or why not?
We return to the notion of a function and examine the allowable inputs.
Motivating Questions
- Are there numbers that cannot be plugged into a given function?
- How do we denote the numbers that can be plugged in?
- What are the allowable inputs for our famous functions?
Introduction
We often think about functions as a process which transforms an input into some output. Sometimes that process is known to us (such as when we have a formula for the function) and sometimes that process is unknown to us (such as when we only have a small table of values).
The Domain of a Function
Let be a function from to . The set of possible inputs to is called the domain of .
The set is called the codomain of .
Let be the population of Columbus, OH as a function of the year. According to
Google, the population of Columbus in 1990 was 632,910 and in the year 2010 the
population was 787,033. That means we can say:
What if we were asked to find ?
This question doesn’t really make sense. There were Mound Builder tribes in the area
around the year 1200, but the city of Columbus was not incorporated until the year
1816. We say that is undefined. That is to say, is not in the domain of
.
Let be the function given by . Is there any number that cannot be used as an input
to ?
There is only one number that is not a valid input, . The number can be divided by
any nonzero number. For instance or are perfectly valid outputs. However, if
someone attempted to plug into the formula , they would end up with a
division-by-zero, which is undefined. The number is not in the domain of
.
When we are given a function, sometimes the domain is given to us explicitly. Consider the function for . The phrase “for ” tells us the domain for this function. We may be able to plug any number into the expression , but it’s only when that this gives our function. For instance, , but is undefined.
Sometimes, when we are given a function as a formula, we are not told the domain. In these circumstances we use the implied domain.
Let be a function whose inputs
are real numbers. The implied domain of is the collection of all real numbers for
which is a real number.
Let be the function given by . Find the domain of .
The only information we are given about is the formula for . That means we are
being asked to find the implied domain. Since the square root only exists (as a real
number) when the expression under the root, called the radicand, is non-negative, we
need to ensure that:
The domain is the set of all for which .
Interval Notation
As in the previous example, solutions of inequalities play an important role in expressing the domains of many types of functions. As a standard way of writing these solutions, we rely on interval notation. Interval notation is a short-hand way of representing the intervals as they appear when sketched on a number line. The previous example involved which, when sketched on a number line, is given by
There are five different types of infinite intervals: the first two are closed infinite intervals (which contain their respective endpoint) and the other three are open infinite intervals (which do not contain the endpoint). For a fixed real number , these are:
- (a)
- represents ,
- (b)
- represents ,
- (c)
- represents ,
- (d)
- represents , and
- (e)
- represents all real numbers.
The notation uses the square bracket to indicate that the endpoint is included and the round parenthesis to indicate that the endpoint is not included.
Not every interval is infinite, however. Consider the interval in the following sketch
For a bounded intervals (ones that are not infinite), there are also four possibilities. For and both fixed real numbers, these are:
- (a)
- represents ,
- (b)
- represents ,
- (c)
- represents and
- (d)
- represents .
Practically, this amounts to writing the left-hand endpoint, the right-hand endpoint, then indicating which endpoints are included in the interval.
Metacognitive
Moment When neither endpoint is included, can be mistaken for a point on a
graph. You will need to use the context to know which is meant.
Finally, isolated points are not included in intervals, but are written in the
form , and multiple disjoint intervals are connected using the Union symbol
.
A piece of wire, 10 meters in length, is folded into a rectangle. Call the width of the
rectangle, as in the image below, and call the height. 
Find a formula for the height as a function of , . What is the domain of
?
The wire forms the perimeter of the rectangle. Since the wire has length 10 meters, that means the sum of the lengths of all the edges is 10. Thus, , or . Solving this formula for gives:
The function is given by .Any number can be plugged into the formula , but we have to take into account where these quantities came from in the story. The value was a length of a side of a rectangle. That means cannot be negative. For a similar reason, cannot be negative.
If has a value larger than , it would force to be negative, which is impossible. The domain of is .Think about what it would mean for or . The value would correspond to a rectangle
with width zero, and would correspond to a rectangle of height zero (since ). For
convenience, mathematicians often allow rectangles of width zero or height zero. If
you are not comfortable with calling those things rectangles, you can use as your
domain instead.
The Domains of Famous Functions
Earlier you were introduced to the graphs of several “Famous Functions”. We will revisit these functions over and over again throughout our studies. For now, we will formalize what we have seen with their graphs.
- (a)
- The Absolute Value function - We can take the absolute value of any number. The Absolute Value function has domain .
- (b)
- Polynomial functions - We can plug any number into a polynomial. All polynomials have domain .
- (c)
- Rational functions - Remember that a rational function is one that can be written as fraction of two polynomials, with the denominator not the zero polynomial. The domain of a rational function consists of all real numbers for which the denominator is nonzero.
- (d)
- The Square Root function - We can take the square root of any non-negative number. The square root function has domain .
- (e)
- Exponential functions - Exponential functions , for with , have domain .
- (f)
- Logarithms - Logarithms have domain . This is similar to the domain of , except the endpoint is not included.
- (g)
- The Sine function - The sine function has domain .
Spotting Values not in the Domain
Of our list of famous functions, notice that only rational functions, radicals, and logarithms have domain that is not the full set of all real numbers, . When trying to find the domain of a function constructed out of famous functions, this gives us some guidelines to follow. The following list is not exhaustive, but gives a good place to begin.
Strategies for Finding the Domain
Remember that the number zero is neither positive nor negative. The non-negative
numbers are , while the positive numbers are .
- (a)
- The input of an even-index radical must be non-negative.
- (b)
- The input of a logarithm must be positive.
- (c)
- The denominator of a fraction cannot be zero.
- (d)
- The real-world context. If a function has a real-world description, this may add additional restrictions on the input values. (You can see this in Example examp:wirePerimeterDomain above.)
Find the domain of the function
Examine the individual terms. The first term is an absolute value function, while the
second and third terms are polynomials. There is no restriction on their domain. The
last two terms, however, are a fraction and a logarithm.
The denominator of the fraction cannot be zero, so
The input to the logarithm must be positive, so
In order for a number to be in the domain of the function, it must be in the domain of every term of the function. That means it must satisfy both and . Altogether, this means the domain is .
Find the domain of the function
The denominator of this fraction is . The graph of is an upward-opening parabola
with vertex at the point . As such, the denominator does not have zero as an output.
Our only restrictions will come from the numerator.
The input to the logarithm must be positive, so
That inequality has graph given by
The radicand must be non-negative, so
That inequality has graph given by
In order for a number to be in the domain of this function, satisfy both and . The points satisfying both inequalities are given in the graph found by overlaying both graphs above
and taking the graph of all points on both graphs.
Altogether, this means the domain is
Piecewise Defined Functions and Restricted Domains
Consider the function for , and the function (given without this restriction). The implied domain of is , but what can we say about ? The formula makes sense when , but the function definition for has the added statement “for ”. This is telling us the domain of is . In this case is undefined.
We can think of the function as coming from the function by deciding that some inputs are not valid. We have restricted the domain.
Suppose we have a function given by for (which has domain ) and a different function given by for (which has domain ). If we are given an -value in the interval , that input can only be plugged into one of these two functions. Let’s create a new function by setting if and by setting if . As a compact way of writing this, we would say:
A piecewise defined function is a function that is given by different formulas for
different intervals in its domain. This is sometimes shortened to just piecewise
function.
The function above is a piecewise defined function. On the interval it is given by the formula , and on the interval it is given by the formula . It has two pieces, one piece is quadratic and the other piece is linear. The graph of the function is given below.
Let be the piecewise defined function given by
What is the domain of ? Evaluate the following:
- (a)
- (b)
- (c)
- (d)
- (e)
The function is given as a piecewise defined function with three pieces. The first piece is used , the second piece is used when , and the third piece is used when . This function is defined for all numbers except those between 3 and 4.
The domain of this function is .
- (a)
- Since , this uses the first piece of the function, so .
- (b)
- Since , .
- (c)
- is between and (it’s approximately 1.57), so .
- (d)
- is not in the domain of , so is undefined.
- (e)
- Since , .
Write the absolute value function as a piecewise defined function.
Do you notice that this graph looks like two straight lines, meeting at the origin?
Let’s focus on the right-hand side first. For , this is a line with slope and -intercept
at the origin . This line has equation . For , this is a line with slope and -intercept at
the origin . This line has equation .
Let’s examine the graph of .
That means agrees with if , and agrees with if . Putting these together gives us:
This formula tells us that the absolute value of a positive number is itself, while the absolute value of a negative number changes the sign.