Characteristics of Function Graphs
Objective
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:
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, +∞).
So what does it mean if a function is increasing?
So, the official definition is: “A function is increasing on an interval if ƒ(x1) < ƒ(x2) when x1 < x2 for any x-values x1 and x2 from the interval.”
Now, in plain English, it means that a function is increasing from one point to another, if the x’s are increasing and y’s are increasing at the same time on a given interval.
It’s a little hard to explain, so here’s an example:
Increasing function
Here’s the function:
As we can see, it looks like as:
x increases
y increases (or the graph goes up)
Now, we need an interval to tell whether or not this function can be called increasing or not, since the definition specifies on a given interval. �
So what does it mean if a function is decreasing?
So, the official definition is: “A function is decreasing on an interval if ƒ(x1) > ƒ(x2) when x1 < x2 for any x-values x1 and x2 from the interval.”
So basically this means that a function is decreasing from one point to another, if the x’s are increasing and y’s are decreasing at the same time on a given interval.
So it’s very similar to what we saw before, just backwards.
Decreasing function
We’re going to use the same function as we did before.
�
As we can see, it looks like as:
x increases
y decreases (or the graph goes down)
LOCAL MAXIMUM AND LOCAL MINIMUM OF A FUNCTION
A local maximum is a maximum that can be seen given a specific interval.
Like the graph we saw before, some graphs have what we call valleys (or low points) and peaks (or high points).
These highs and lows are considered the local maximums and local minimums of a function.
The peaks are the maximums, and the valleys are the minimum.
So why do we call them local?
Because they are specific to the interval given.
The actual maximum of a function is usually referred to as the global maximum, and is the highest point of the function at all intervals. �Or, in other words, the highest point of the entire function.
So let’s look at a quick example to tell the difference.
THE LOCAL MAXIMUM
Let’s use the same graph (because I’m lazy and don’t want to go making another one).
Again, it’s important to point out that the local maximum is not the highest point of the entire graph, it’s just the highest point of that interval.
THE LOCAL MINIMUM
Now let’s talk about the local minimum.
The local minimum is the same as the local maximum, except that it’s the lowest point of the graph.
So, take the same graph (again, lazy!).
The zeroes of a function
Finally, the zeroes of a function are where the graph crosses the x-axis.
This only occurs when you set y = 0.
They are also fairly easy to spot, so let’s look at an example:
As we can see from this graph, at (0,0) and at (3,0) it crosses/touches the x-axis.
So the zeroes of this graph would be: x = 0, x = 3.
And yes, that’s how you would write that.
Example:
As we can see, it seems like the graph is decreasing from -4 to -3, but then drastically increases from -3 on, so we would say this is locally increasing.
The local maximum is going to be at y = 9 since that’s the highest point of the graph, and the local minimum would be at y = 0, the lowest point of the graph.
The zero of the function (given the interval) would be at x = -3
Example:
As we can see, from 0 to 3, as x increases, y decreases, so we would say at this interval, the function is decreasing.
The local maximum is going to be at y = 9 since that’s the highest point of the graph, and the local minimum would be y = 0, at the lowest point.
The zero of the function (given the interval) would be at x = 3
Example:
As we can see, from 4 to 8, as x increases, y increases; so we would say this graph increases.
The local maximum is going to be at y = 7 since that’s the highest point of the graph, and the local minimum would be at y = -1, the lowest point of the graph.
The zero of the function (given the interval) would be at x = 5