Domain & Range of Functions
Objective
Definitions to go over
Interval – an interval is part of a number line without any breaks in it. So think of a solid line segment but placed on a number line. There are two types of intervals:
Finite interval – has two end points which might be included in the interval, or they might not be included (depending on how it is described).
Infinite Interval – is considered unbounded at one, or sometimes both ends
Now there are a few ways to write what an interval is, so let’s get started:
HOW TO WRITE A DESCRIPTION OF AN INTERVAL:
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So, what does this all mean?
Well, for set notation, the vertical bar means “such that,” so you read {x|x ≥ 1} as “the set of real numbers x such that x is greater than or equal to 1.”
All real numbers from a to b, including a and b
Description of Interval
Type of interval
Inequality
Set Notation
Interval Notation
Finite
[a , b]
All real numbers greater than a
Infinite
All real numbers less than or equal to a
Infinite
For interval notation, it’s important to note that a square bracket indicates that an interval includes an endpoint, but a parenthesis indicates that an interval doesn’t include an endpoint.
For an interval that is unbounded at its positive end, use the symbol for positive infinity, +∞.
For an interval that is unbounded at its negative end, use the symbol for negative infinity, -∞.
Always use a parenthesis with positive or negative infinity, never a square bracket.
So, you can write the interval x ≥ 1 as [1, +∞).
Example
Complete the table by writing the infinite interval shown on each number line as an inequality, using set notation, and using interval notation
Infinite Interval | | |
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Inequality
Set Notation
Interval Notation
SO WHAT ABOUT INFINITE INTERVALS NOT BOUNDED BY BOTH SIDES?
So far we’ve only been dealing with infinite intervals that are bounded by one side (they have a starting point), however what about infinite intervals that are truly unbounded?
Well, here’s an example:
We can see that this is an infinite interval because the arrows point to both sides.
This means that the interval is unbounded, meaning it doesn’t stop.
The way we write this is as such:
Interval Notation:
Set Notation:
It’s also important to note that the numbers included in this interval would be: All real numbers.
IDENTIFYING A FUNCTIONS END BEHAVIOR
Recall that the domain of a function f is the set of input values x, and the range is the set of output values ƒ(x).
What we’re talking about though, is the end behavior, which describes what happens to the ƒ(x)-values as the x-values either increase without bound (approach positive infinity) or decrease without bound (approach negative infinity).
For example, if we look at this graph, we can say that:
Statement of End Behavior | Symbolic Form of Statement |
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As the x-values increase without bound, the f(x)-values also increase without bound.
As the x-values decrease without bound, the f(x)-values also decrease without bound.
THE DOMAIN OF A FUNCTION
Now that we understand how to represent an interval on a Number line, let’s talk about Domain.
So, for the official definition, the Domain of a Function, is all of the possible values of x.
Now, for some graphs, there is a limit on what x can be, but those are usually stated before hand.
For most other’s, there is no limit, because the function is unbounded.
For example:
If we look only at the x-axis of this graph, we can see that this graph has no limit on what x can be.
Therefore, we would say that:
Domain:
Inequality:
Set Notation:
Interval Notation:
However, if we are given a graph such as:
We would then see that:
THE RANGE OF A FUNCTION
Now that we understand how to identify the domain of a function, let’s talk about Range.
So, for the official definition, the Range of a Function, is all of the possible values of y.
Now, for some graphs, there is a limit on what y can be, but those are usually stated before hand.
However, it is important to note, they are more common than having a limited domain.
For most other’s, like with domain, there is no limit, because the function is unbounded.
For example, looking at our past example:
If we look only at the y-axis of this graph, we can see that y must be greater than or equal to zero.
Therefore, we would say that:
Range:
Inequality:
Set Notation:
Interval Notation:
However, if we are given a graph such as:
We would then see that:
SO, SINCE WE NOW REMEMBER DOMAIN AND RANGE, LET’S TRY IT OUT:
Identify the domain and range of the following:
So, first thing we do is graph the equation:
So now that we can see the graph, we can examine the x-coordinate behavior (or the domain).
As we can see, it doesn’t look like the graph has any boundaries.
So, for the domain, it’s safe to say:
Inequality:
Set Notation:
Interval Notation:
Now, let’s look at the y-coordinate behavior (or the range)
As we can see, it doesn’t look like the graph has any boundaries.
So, for the range, it’s safe to say:
Domain:
Range:
EXAMPLE 2
Identify the domain and range of the following:
So now that we can see the graph, we can examine the x-coordinate behavior (or the domain).
As we can see, it doesn’t look like the graph has any boundaries.
So, for the domain, it’s safe to say:
Inequality:
Set Notation:
Interval Notation:
Now, let’s look at the y-coordinate behavior (or the range)
As we can see, the graph has a boundary at around -2.25.
So, for the range, it’s safe to say:
Domain:
Range:
So, first thing we do is graph the equation:
EXAMPLE 3
Identify the domain and range of the following:
So now that we can see the graph, we can examine the x-coordinate behavior (or the domain).
As we can see, it doesn’t look like the graph has any boundaries.
So, for the domain, it’s safe to say:
Inequality:
Set Notation:
Interval Notation:
Now, let’s look at the y-coordinate behavior (or the range)
As we can see, the graph bends, but never touches y = 0.
So, for the range, it’s safe to say:
Domain:
Range:
So, first thing we do is graph the equation: