okay hey Jonathan here at Colfax math I'm gonna go over finding inverses and I
have a five-step process to do that most important thing to note really is f of X
is your Y value so f (x) and y are interchangeable F to the negative 1 of X
that is actually not to the power negative 1 it is saying it's an inverse
function so this is your inverse function so that's the notation to say
it's the inverse function okay I have a five step process step one is replace f
of X with y step two is switch your X's and Y's three is going
to be solve for y step four is replace your new Y with
the function notation inverse function of X and then step 5 is plug in the
value if you're given one okay so this right here would be kind of a common
problem right here it's really a two-step problem
step one is find the inverse function so it's really saying what is the inverse
function of X and then step two is once you find that inverse function take 15
and plug it in okay so here I am with red here so first thing I'm going to do
is replace f of X with y so I'm going to take that and replace it with a y so
that f of X just becomes y is equal to 6 plus 3 the square root of 1 minus 2x
step tu is saying switch the X and the y so take that X put it there
take that Y and put it there so that now gives me X is equal to 6 plus 3 square
root of 1 minus 2 y so that's step 2 you now I need to solve for y I'm going to
subtract 6 from both sides giving me X minus 6 equals 3 times that I'll divide
both sides by 3 to get the square root of 1 minus 2 y still solving for y I'm
going to square both sides that whole thing is squared square and square root
will cancel now I have 1 minus 2 y is equal to the quantity X minus 6 over 3
squared subtract 1 from both sides that'll cancel have to get rid of that
negative 2 divide both sides by negative 2 and now I have Y by itself I know it's
a big ugly thing but it's probably just as good to leave it that way so now I
let's see let me run through my steps again replace f of X with y switch my
X's and Y's solve for y so that's what I did there so this is solving for y right
y is now equal to that now step 4 replace my new Y with that function
notation so replace this new Y with inverse function of X and that's going
to be equal to the quantity X minus 6 over 3 squared minus 1 all divided by
negative 2 then finally step 5 is plug in the value of 15 so usually
Part B of the problem I'm going to take that value of 15 and plug it in I take
that value of 15 and plug it in there okay so finally
step 5 is find the inverse function of 15 so I found the inverse function now
take 15 and plug it in there so I have 15 minus 6 divided by 3 quantity squared
minus 1 all divided by a negative 2 15 minus 6 9 divided by 3 is 3 squared is 9
minus 1 is 8 so this is going to be a 8 in the numerator over negative 2 which
is equal a negative 4 so again this is how to find the inverse function and
then once you have the universe function plug in a value into the inverse
function and this is my five step process so hopefully that helped with
finding inverse functions and plugging in values into inverse functions thanks
for watching