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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
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This section covers Adding/Subtracting/Multiplying/Dividing
Functions, Increasing, Decreasing and Constant Functions, Even and Odd
Functions, and Compositions of Functions, including how to Decompose
Functions, Domains of Composites, and Applications of Compositions.
Now that we know what a function is from the Introduction to Functions
section, let’s talk about some cool things that we can do with them, and talk
about more advanced types of functions. All these types of things are found in
“real life” computations that go on every day!
Let’s first review what a function is. A function is defined as a relation between
two things where there is only one “answer” for every “question”. In a function,
the x is the “question” and the y or f(x) is the answer. So if we have a function
, if our x was 2, our f(x) would be
Also,
So with functions, just remember, for the “x” on the left, plug this value into
every “x” on the right.
Adding, Subtracting, Multiplying, and Dividing Functions
Functions, like numbers, can actually be added, subtracted, multiplied and even
divided. These operations are pretty obvious, but I will give examples anyway.
Let’s think of a situation where two functions would be added:
Let’s say yesterday you went to your favorite cosmetics store and used your $5
off coupon. Then you went to your favorite shoe store and used your 25% off
coupon. What is the function for the total number of dollars you spent
yesterday, given you bought items that regularly cost $80 at each store?
(Disregard tax). (I know – makeup is really expensive these days!)
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
So we have two functions here: one for what you spent at the cosmetic store
(given the regular price of the items), and one for what you spent at the shoe
store (given the regular price of the items). You might want to put some “fake”
numbers in to convince yourself that these are the correct functions.
Let’s get the total amount you spent two ways: first adding the amount you
spent separately at each store, and then combining (adding) the functions and
getting the total amount you spent at both stores.
We said that you bought items at each store that regularly cost $80. (There’s a
lot you can buy at cosmetics stores these days!). Look how we get the same
amount that we spent by first evaluating the functions separately (plugging
numbers in) and then adding the amounts, and second by adding the functions
and then evaluating that combined function:
So below are the rules on how the adding, subtracting, multiplying and
dividing functions work. I know the notation on the left looks really funny (and
we saw this in the example above); it just means that the
sum/difference/product/quotient of two functions is defined as when you just
take the right hand side (what they are defined as) and
add/subtract/multiply/divide them together. Makes sense, right?
Let’s use two different functions, both of which are binomials:
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
See – not too bad, right? I know we haven’t learned about the quotient of two
functions yet; we will talk about that in the Rational Functions section. This
particular function that we got from dividing functions can’t be simplified, but
we’ll see later that some of them can.
Note that the multiplication operation on functions is not to be confused with the
composition of functions, which looks like
. We will go over this later
here.
Increasing, Decreasing and Constant Functions:
You might be asked to tell what parts of a function are increasing, decreasing, or
constant.
This really isn’t too difficult, but you have to be careful to look where the y or
f(x) is increasing, decreasing, or remaining constant, but the answer will be in an
interval of the x. The answer will always with soft brackets, since the exact
point where the function changes direction is neither increasing, decreasing, nor
remaining constant.
So let’s look at an example:
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
See how we are looking at the y (up and down) to see where the function is
increasing, decreasing, or constant, but write down the x (back and forth)?
Also, see how the function is either increasing, decreasing, or constant for its
whole domain (except for the “turning” points)?
Also, note in the example above that there is a local maximum at point (3, 2)
since this is the highest point in the section of the graph where the “hill” is? A
local minimum would similarly be the lowest point in a section of a graph where
there is a “valley”. This graph would have no absolute minimums or absolute
maximums (the absolute lowest and highest points on the graph), since the
range of the graph is
.
Even and Odd Functions:
There are actually three different types of functions: even, odd, or neither.
Most functions are neither, but you’ll need to know how to identify the even and
odd functions, both graphically and algebraically. One reason the engineers out
there need to know if functions are even or odd is that they can do less
computations if they know functions have certain traits.
Even Functions
Even functions are those that are symmetrical about the y axis, meaning that
they are exactly the same to the right of the y axis as they are to the left. This
means if you drew a function on a piece of paper and folded that paper where
the y axis is, the two sides of the function would match exactly. (You will go over
all this symmetry stuff in Geometry). One of the most “famous” examples of an
even function is
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
I know this sounds really complicated but this means if (x, y) is a point on the
function (graph), then so is (–x, y). But the most important way to see if a
function is even is to see algebraically if
We’ll see examples
soon!
Odd Functions
Odd functions are those that are symmetrical about the origin (0, 0), meaning
that if (x, y) is a point on the function (graph), then so is (–x, –y). Think of odd
functions as having the “pinwheel” effect (if you’ve heard of a pinwheel); if you
put a pin through the origin and rotated the function around half-way, you’d see
the same function. One of the most “famous” examples of an even function is
But the most important way to see if a function is odd is to see algebraically if
“Neither” Functions
Any function that isn’t odd or even, is (you guessed it!) neither!
Here are some examples with some simple functions:
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
Here are more examples where we’ll determine the type of function (even, odd,
or neither) algebraically. Make sure to be really careful with the signs,
especially with the even and odd exponents.
HINT: when dealing with polynomials, note that the even functions have all zero
or even exponents terms – don’t forget that the exponent of a constant, like 8,
is 0, since 8 is the same as
, or (8)(1). Odd functions have all odd
exponents terms (note that x has an exponent of 1). Functions that are neither
even or odd have a combination of even exponents and odd exponents terms.
Note that this works on polynomials only; for example, it does not necessarily
work with a function that is a quotient of two polynomials.
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
Compositions of Functions:
Compositions of functions can be confusing and a lot of people freak out with
them, but they really aren’t that bad if you learn a few tricks.
Composition of functions is just combining 2 or more functions, but evaluating
them in a certain order. It’s almost like one is inside the other – you always
work with one first, and then the other. That’s all it is!
Let’s start out with an example (shopping, of course!). Let’s say you found two
coupons for your favorite clothing store: one that is a 20% discount, and another
one that is $10 off. The store allows you to use both of them, in any order. You
need to figure out which way is the better deal.
Let’s first define the two functions. Make sure you understand how they work
with “real” numbers with each discount, for example if you bought $100 worth of
clothes (you’d spend $80 with first discount, $90 with second). Of course, if you
could only use one of the discounts, the amount you save depends on how much
you spend, but you are allowed to use both.
Let’s look at what happens when we apply the discounts in different orders, if we
were to buy $100 worth of clothes. For now, look at first two columns only:
Now let’s look at the last column. Composition of functions are either written
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or
. The trick with compositions of functions is that we
always work from the inside out with
or right to left with
.
You look at what the inside (or rightmost) function first and then that’s the “new
x“ that we use to replace every “x“ in the outside (or leftmost) function.
Another way to look at it is the output of the inner function becomes the input
of the outer function.
So, in the example above, if we take the $10 off first and then take 20% off, we
have to use the inside function g(x) “x – 10” and put it in the outside function
f(x) as the “new x” and put this everywhere on the right of the f(x) function.
Don’t worry if you don’t get this at first; it’s a difficult concept!
So it’d be better to take the 20% off first (which makes sense, since we’re taking
the 20% off of a higher number).
Compositions Algebraically
So let’s do more examples of getting compositions of functions algebraically.
Let’s use the two functions:
Compositions Graphically
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
Sometimes you’ll be asked to work with compositions of functions graphically,
both “forwards” and “backwards”. You can do this without even knowing what
the functions are!
Let’s first do a “real life” example:
Let’s say you took a very part-time babysitting job (at $12/job) to satisfy your
cravings for going to the movies (at $10 per movie). How much money you make
for the month depends on how many babysitting jobs you get that month
and how many movies you go to depends on how much money you have that
month
So do you see how many movies you go to depends on how many
babysitting jobs you have that month
(Remember, we work with
inside functions first.)
Let’s show this graphically:
So let’s say we took 5 babysitting jobs last month. That translates into $60 of
babysitting money that month (see graph to left). That output of $60 then is
used as the input into the graph at the right. The output of the new graph is 6.
So we could go to 6 movies that month.
To see this algebraically,
Here are more examples, with more complicated graphs:
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I know this can be really confusing, but if you go through it a few times, it gets
easier!
Decomposition of Functions
Sometimes we have to decompose functions or “pull them apart”. I like to think
of this like performing surgery on them – opening them up and seeing what the
pieces are!
Note that there may be more than one way to decompose functions, and some
functions can’t even really be decomposed! But you should get problems that
make it pretty straightforward.
The best way to show this is with an example. Let’s say we have the following
functions:
Compose Function:
Now, let’s first figure out what
is (compose the
function). I know this looks really difficult, but if we just take one step at time,
it’s not too bad. To “build” the function, we need to start from the
inside; normally, we would simplify by multiplying out and combining like terms,
but we’ll be lazy for now .
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Decompose Function:
Now, let’s say we were given
(notice that I didn’t
simplify totally, because this would be too crazy difficult!).
Let’s go the opposite way, or “decompose” the function: look at the last
operation done, and that will be on the outside of the composed function.
So the last thing we did was add 3 (f(x)), before that, we took the square and
subtracted 4 (g(x)), and even before that, we doubled (h(x)) “5x + 8” (k(x)) to
get “10x + 16”.
When we decompose, the function on the outside (or the left) is the last thing
we do. Since we performed the functions above in the following order: k(x) first,
followed by h(x), then g(x), and finally f(x), we have to write the composition in
the opposite order. So
Tricky!
Let’s try another one; decompose
functions again:
. Let’s look at the
The last thing we did was square something and subtract 4 (g(x)), before that,
we took something and multiplied it by 5 and then added 8 to it (k(x)), before
that, we added 3 to something (f(x)), and then before that multiplied x by 2
(h(x)). Again, since we work from the inside out, we have to start with the last
thing we did and go forward:
These are like fun puzzles (sorry – the nerd in me is coming out)!
Domains of Composites Algebraically
Sometimes in Advanced Algebra or even Pre-Calculus you’ll be asked to find the
domains of compositions of functions, both graphically and algebraically. This
isn’t totally intuitive, but if you learn a few rules, it’s really not bad at all.
Let’s first review the case where you have to worry about domains; we looked at
it here in the Introduction to Functions section.
Again, we know so far that domain is restricted if: it is indicated that way in
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
the problem, and/or there is a variable in the denominator and that
denominator could be 0 and/or there is a variable underneath an even
radical sign, and that radicand (underneath the radical sign) could be
negative.
The best way to show how to get the domain of a composite is to jump right in
and do a problem. Suppose
and
. Let’s find
the composition
and it’s domain.
To get the composition, we put the inner function as the “new x” in the outer
function, so
.
It’s interesting to note that we can’t just look at the composite function to get
the domain of the composition of functions; unfortunately, it’s a little trickier
than this. We first need to see what values x can be for the inside function,
since x goes directly into that function.
Next, since the output (the range) of the inside function goes into the input (the
domain) of the outside function, we have to make sure that the inside function
works for the outside domain. But we still need to solve for x to get this
restriction.
Then we have to put these two restrictions together to get the domain of the
composite. Since all this is very confusing, I like to just remember the following
“trick”:
Domain of Composite = The Intersection of {the Domain of the Inner Function}
and {Restricting the Inner Function to the Outer Domain}. Another way to write
this is the Domain of the Composite is:
.
Now I know this is really confusing, so let’s work the problem with
and
to find the domain of the composite
.
The domain of the inner function f(x) is
, since that’s what’s given. Now
let’s put the inner function as the “new x” in the outer function and get the
domain. So we have to get the domain of
. So
, since what’s underneath an even root has to be 0 or positive.
See how we had to solve for x again?
(We could have also seen that the domain of the outer is
because of the
even radical, and then gotten the inner function (the “new x”) in the outer
domain this way:
.)
Now we have to take the intersection or “and” (both have to work) of
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it’s not both
, which is
. (See that, for example, .5 doesn’t work, since
0 and 1). So the domain of
. Whew – that
was difficult! These take a lot of practice!
Remember also that if either the domain of the inner or domain of outer (where
you put the inner) is all real numbers (no restrictions), you don’t have to worry
about that part of the intersection (see examples below).
Let’s do more examples. For the following functions, find the composition and its
domain:
Let’s do one more that happens to involve a Rational Function, that we will
learn about here in the Solving Rational Functions, including
Asymptotes section. To get the domain of the following composition, we will
have to use a sign chart.
We’ve worked with sign charts here in the Quadratic Inequalities section, and
they are a useful tool!
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Domains of Composites Graphically
We can determine the domains of composites graphically too, and this way is
actually better to see what’s going on. Here are some examples:
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Compositions of Functions, Even and Odd, and Increasing and Decreasing - She Loves Math
If we knew what these functions were algebraically (we’ll see how later!), we
could use the
method.
I know this is really difficult, but follow these steps and you’ll be able to do any
of the problems!
Applications of Compositions
Here are a couple more composites applications that you may see in your
Algebra class:
Increasing Area Problem:
A rock is thrown in a pond, and the radius of the ripple circles increase at a rate
of .5 inch per second. Find an algebraic expression for the area of the ripple in
terms of time t, and find the area after 20 seconds.
Solution:
For these types of problems, we want to start with the simplest (meaning most
direct) function we can find in terms of time or t; this would be that the radius of
the ripples increase at a rate of .5 inch per second. We can write this as
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. Now we also now that the formula for the area of a circle based on its
radius is
.
Since the area is based on the radius which is based on the time, we can put
the two functions together, with the inside function being the time
function (remember that we work from the inside out with compositions).
We get
. For when t = 20 seconds (make sure units
match, which they do since we are dealing with a rate of .5 inch per second!), we
get
.
Shadow Problem:
Amelia is walking away from a 20 feet high street lamp at a rate of 4 ft/sec. If
Amelia is 5’6″ tall, how long will she have walked (both in terms of time and
distance) when her shadow is 7 feet long?
Solution:
This is a tough one; actually you’ll be doing some like this in Calculus. Let’s draw
a picture first, and we need to know a little bit about shadows and Geometry.
Learn these rules, and practice, practice, practice!
On to Inverses of Functions – you are ready!!
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