Section 1.7: Analyzing Graphs of Functions and Piece-wise Defined Functions
§1 Even and Odd Functions
When we say even and odd functions, we really mean the symmetry of the function. An even function is
symmetric with respect to the y-axis. An odd function is symmetric with respect to the origin. We can
also determine if a function is even or odd algebraically. If the function is not even and not odd, then we
just say that’s its neither.
A function is even if, for every number x in its domain, the number –x is also in the domain and
f (− x) =
f ( x)
A function is odd if, for every number x in its domain, the number –x is also in the domain and
f (− x) =
− f ( x)
The definition seems a little confusing, but it basically means we replace every x in the function with a –x.
Then you simplify the function. If the result is the same as the original function, then the function is even.
If the result is the negative of the original function, then the function is odd. Let’s try a couple examples.
Determine if the following is even or odd: f (=
x) 2 x 2 + 1 . So the first thing we do is replace every x
with –x. You should end up with f (− x) = 2(− x) 2 + 1 . When we simplify this, we end up with
f (− x) = 2 x 2 + 1 . But this is just the same as f ( x)! Hence since f (− x) =
f ( x), the function is even.
Remember, an even function is symmetric with respect to the y-axis.
Determine if the following is even or odd: f ( x) =
− x 3 + 2 x . First find f (− x). You should end up with
f (− x) =−(− x)3 + 2(− x) . Once we simplify this, we end up with f (− x) = x 3 − 2 x . Note, however,
that this is just equal to − f ( x)! Hence this function is odd, which means it is symmetric with respect to
the origin.
PRACTICE
1) Determine if the following function is even or odd or neither: f ( x) =
− x2 − 2x − 6
2) Determine if the following function is even or odd or neither: f (=
x) 2 x3 − x5
§2 Piecewise Defined Functions
A piecewise function is a function that has one or more different equations on different parts of its
domain. For example, try sketching the following piecewise function:
 x2 , x ≤ 0
f ( x) = 
 x + 1, x > 0
Note that for this function we have two different functions, one quadratic and one linear. Also note that
we have restricted the domain for both. Pay particular attention to the endpoints – make sure you
denote which endpoint is included for its respective function using closed and open circles.
Say for the previous function, I ask you to evaluate f (−4), f (0), and f (3) . You need to use the
correct ‘piece’ of the function in order to evaluate the values properly.
PRACTICE
3)
§3 Increasing, Decreasing, and Constant Functions
Once again, what does this mean? Basically, the graph of an increasing function goes up from left to
right; the graph of a decreasing function goes down from left to right, and the graph of a constant
function remains at a fixed height.
To answer the question of where a function is increasing or decreasing or constant, we use open
intervals of the x-coordinates.
For the graph above, the function is increasing in the intervals ( −2, 0 ) and ( 2, 4 ) . It is decreasing in
the intervals ( −4, −2 ) and ( 0, 2 ) .
PRACTICE
4) Determine the intervals where the function is increasing, decreasing, and/or constant:
§4 Local and Absolute Maxima and Minima
I won’t give the book definition yet because it gets a little too technical – we’ll get into the book
definition a little later. The maxima and minima of a graph refers to the highest and lowest points of the
graph. Local maxima and minima refers to a certain open interval within the graph. The absolute
maxima and minima refers to the whole domain of the graph. So you can ask this question to find local
maxima or minima – within a certain interval within the graph, is there a point that is higher or lower
than every other point? Look at the following graph:
We can see that the domain of this function is all real numbers. But are there any local maxima or
minima? We can see that there are two local minima and one local maximum. These are points within a
small interval of the whole domain that are the lowest point (for the minima) of that interval, and the
highest point (for the maxima) of that interval. We see that the two local minima are 1 and 0. Note that
when we say what the minima are, we give the function or y-values. We say that the minima occur at x =
-1 and x = 3. The local maximum is 2, and it occurs at x = 1.
Absolute maxima and minima are the lowest and highest values within the whole domain. Make sure
you examine the graphs carefully to determine what the domain is. Arrows at the end of the graph
mean that the graph extends in that direction – it doesn’t simply stop at that point.
PRACTICE
5) Determine the local and absolute maxima and minima for the following:
§5 More on Local and Absolute Maxima and Minima
Let’s look at what the book says about local max and min:
This is what the book says about absolute max and min:
As you can see, seems a little bit confusing and wordy. Let’s try to think about it another way.
§6 Absolute Max and Absolute Min
Given the graph of a function, we should be able to determine what the domain and the range is. The
absolute max is the single largest y-value on the graph. The absolute min is the single smallest y-value
on the graph. Note that the absolute max and min are real numbers. So infiniti does not count! Hence a
function may not always have an absolute max and/or absolute min.
Think about it another way. The range of the function actually gives you the endpoints of the min and
max. If one of these endpoints is infiniti, than it will not be one of the extrema.
Look at the following:
What is the range of the function? We can see that the range is [ 0, ∞ ) . Note that the range tells us
what the absolute max and min are. Remember, though, that infiniti does not count, since it is not a real
number. Hence, the absolute min is 0, and there is no absolute max!
Look at another example:
The range is [ 0, 4] . We can see on the graph that the single smallest y-value is 0, which occurs at x = 5.
Hence the absolute min is 0. The single largest y-value on the graph is 4. Hence the absolute max is 4.
§7 Local Max and Min
This gets a little more tricky. The local max and min refer to an open interval within the domain. So, is
there a y-value (within an open interval) that is bigger (or smaller) than the ones around it?
Since the definition says that the interval must be open, we DO NOT include endpoints as local max or
min, although they may be an absolute max or min. The easiest way on a graph to tell where the local
max and min are is to look for any valleys or hills.
Hence it is possible for a local max to be an absolute max, and for a local min to be an absolute min. I
always suggest finding the local extrema first, then the absolute extrema.
Look at the following:
Note that the range is [1, 4] .
To find the local min, look for a valley. Note that at the point (1,1), it is the lowest point in relation to the
graph around it. Hence, we can say that there is a local min at 1.
Similarly, there is a hill at (3,4). So we can say that there is a local max of 4 at x = 3. There are no other
hills or valleys in the graph, so there are no other local extrema.
Once again, note that we do not count the endpoints as local extrema!
What is the absolute max? The single largest y-value is 4.
What is the absolute min? The single smallest y-value is 1.
Note that in this example, the local max and absolute max are the same, as are the local and absolute
min!
Another example:
The range is [ −1, ∞ ] .
First find the local max. Is there are any hill on the graph? Yes, at (0,1). Hence there is local max of 1 and
x = 0. There are no more hills, so no other local max.
Now find the local min. Are there any valleys on the graph? There are two, but in this case they are the
same. Hence the local min is -1, which occurs at x = π and x = −π .
Is there an absolute max? Well the range is [ −1, ∞ ] . Hence there is no single largest value of y. There is
no absolute max.
Is there an absolute min? The single smallest y-value is -1.