Composition of Functions lesson 3
Dec 4, 2023
http://www.moomoomath.com/Inverse-Functions.html
Composition of Functions Lesson 3
0:50 Evaluating Functions
1:00 Solves g(2)
1:45 Solves composite function ex.2
3:35 Solves the function h(g(x))
5:45 Finding the functions
5:54 Finds the function g(h(x))
7:18 Finds the function h(g(x))
7:58 Finds the function f(j(x))
9:00 Finds the function j(f(x)
This is video two in a series of videos covering exponents and functions. Here is the complete list of videos:
Lesson 1 Properties of Exponents
http://youtu.be/Z6qmnNW7WCk
Lesson 2 Function Operation
http://youtu.be/OKWrifSxPSk
Lesson 3 Composition of Functions
http://youtu.be/CQADqGkY2wU
Lesson 5 Finding the inverse function
http://youtu.be/HKFccjtYYq8
Lesson 6 Verifying Inverse Functions
http://youtu.be/cmieFR_3bnQ
Lesson 7 Inverse Functions and Graphs
http://youtu.be/iiankYTC1zM
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welcome back to lesson three composition
of functions
in the unit
composition and inverse of functions
today today we're learning something
probably that's totally new to you and
that's composition of functions the
other two lessons might have been a
little bit of review but this one's
probably totally new to you okay
hopefully you've printed your note
taking guide and i've got one two three
four functions at the top f of x g of x
h of x and j of x four different
functions we're gonna use to work
through some examples okay so part one
we're just gonna talk about how do you
evaluate a fraction uh function well you
guys have evaluated one function before
so let's just start with that
g of two okay to evaluate a function
like g of two we would go up figure out
which one's the g function and just plug
in a 2 for x so let's go up here and
we're going to take the g
of x function which is 2x minus 3
and for x we're going to plug in a 2
so everywhere there's an x we're going
to plug in 2 so that's going to be 2
times 2 which is 4 4 minus 3 is positive
1. so the answer to that would be just g
of 2
is equal to a positive 1. okay so that's
evaluating one function but what happens
when we have this
composite function so we need to look at
this notation to understand what it
means okay composite functions are when
one function is inside another function
so what this is telling us is we're
going to evaluate the g function for 2
which is what we just did and then take
that answer and plug it into the h
function so you kind of work from the
inside out okay so g of 2 is our first
step once we get that we take that
solution and plug it into the h function
okay so we're starting with the option
one which is just to evaluate one
function at a time so i'm going to take
the answer that we just got which is a
positive 1 and i'm going to plug it into
the h function so let's go up here
identify the h function
so we figured out that g of 2 is equal
to 1. now we're going to do g of
h of g of 2. so we're going to take the
answer to that
and we're going to
evaluate h of 1
okay
so
each function is
x
minus 2
and we're going to evaluate it for that
one that we just got out we're going to
plug it in
and get 1 minus 2 which is negative 1.
so the answer would be
h
of g
of 2
is equal to negative 1 and that's
exactly how you read it
h of g of 2
would be equal to negative one so one
inside the other okay so that's one
option you just take the function and
work from the inside out okay option two
if you're doing a composite function you
can actually figure out what that
composite function is first then plug
the value 2 in so let me show you how
that works
okay we're going to take
the same example
h
of g
of x
and evaluate it for h of g
of 2.
but we're going to start first by
finding the composite function so we're
going to take the g function and plug it
into the h function
okay so what i like to do is i actually
like to start with the outer function
first and i like to write this but i
like to leave a blank where the h over
the x is
so my
h function
is equal to x minus 2. well i'm going to
leave a big blank where this x is
and i've got to put the g function in
where that x is my well my g function is
2 x minus 3.
so i have taken my g function and i've
plugged it in for x okay take a second
and digest that i took the g function
and i put it inside the uh h function
everywhere there's an x now i just
simplify it 2x minus 3 minus 2 is minus
5. so that means my h of g
of x function
is 2x minus 5.
now all i do is go up and plug in that
2. now i'm going to evaluate the
composite function
so i'm going to take g
of h of 2
and plug in a 2 everywhere there's an x
and this one only has 1 x so 2 times 2
is 4 and 4 minus
is negative one huh i got the same
answer okay so you've got two options
one i evaluate the function
for the first function
for the value take that solution and
plug it into the second function or
write the functions first and then
evaluate at the end you're going to have
to be able to do it both ways so you
want to make sure you understand both
options
and sometimes it's a little quicker to
to do the first option
okay
so then the second part of today's
lesson we're actually going to do
finding the functions so we're just
finding these composite functions we're
not evaluating it we're just finding
them so we're going to start with
g of h of x okay we started with
h of g of x so we started the opposite
way now let's try it this direction okay
so g is my outer function and i forgot
to plug the h function into it so i'm
going to start by writing my g function
and i'm going to leave a blank wherever
there's an x so let's go up here
and i'm going to have to leave a blank
right here after the 2.
so my g function
is
2 x
minus 3.
now i'm gonna plug the h function in
well what's the h function the h
function is x minus 2. so i'm going to
take that and plug it in to that
function so i'm going to put in an x
minus 2. now notice that's a binomial so
what am i going to have to do i'm going
to have to distribute
so that's going to be 2x minus 4 minus 3
or 2x minus 7. so that means
there's my h function
my g of h of x function
is simplified down to 2x minus 7 okay so
that's how we put together and find that
composite function
now let's
reverse it
let's reverse it
and we're going to have
h
of g of x so at the age function i'm
going to leave a blank where the x is
and then i'm going to plug in my g
function so let's go up here and leave a
blank here
so my h function is my outside function
my g function is my inside function okay
so h is
x minus 2
and then i'm going to plug in 2x minus
3. okay we simplified this a moment ago
so 2x minus 3 minus 2 is 2x minus 5.
okay
now let's try the next one
f of j okay so now we're using two new
functions so f is my outside function
and j is the inside function so let's
start by writing the f function
but we're going to leave a blank right
here an x squared oh that's going to be
tricky because we have a square around
the large function
so f
of j
of x
would be
blank
all squared
plus 8.
okay so our j function is just a
monomial so it's a little bit easier to
plug in so let's plug in a 5x squared in
for that x
5 x squared okay now we have to simplify
that
okay 5x squared quantity
meaning all of it is squared so you have
to square the 5 and square the x squared
so that's going to be 25 x to the fourth
plus eight so there is my composite
function
25 x to the fourth plus eight
okay now let's reverse it
okay we're going to put the f function
inside of the j function so let's write
the j function first leaving a blank and
then put the f function in it
okay so j
of f
of x
is going to be j function is 5
x
squared and we're not squaring the 5
this time only the x because the
remember the f function is 5 x squared
where we're only squaring the x and
we're going to plug into it the f
function which is x squared
plus eight
okay this one is a little trickier
what are we gonna have to do with this x
squared plus eight
we're gonna have to square it out we're
gonna have to foil it
so i'm gonna bring down my five
i'm gonna write x squared plus eight
times x squared plus eight