Huh? How do you solve this? Functional equation for precalculus students

Learn how to solve a function equation based on an inverse function for your precalculus class. We will see how to solve for f(x) if f(3x/(2x+1))=4/(5x-6). We will cover how to find the inverse of a rational function and how to simplify complex fractions.
-----------------------------
-----------------------------
#math #algebra #mathbasics

[justagreekhistorian]

As a 10th grader, I have no idea why I am watching these videos but they are just super addicting

[alibekturashev]

2:29 you can just plug in t since you've already defined it in the Let statement

[Ninja]

We don’t even need to substitute x with the expression for t inside the function because we already said in the very first line that 3x/(2x+1)=t so we know the LHS is f(t) straight away.

Still great video.

[selahattinkara-oh]

Your teaching style and solution method are great.👏

[mauriziomorales]

You are a genius!!! It's the true. Thank you very much.

[davidstigant]

A few weeks ago, I noticed something interesting about inverses of rational functions of the form f(x)=(Ax+B)/(Cx+D) and 2x2 matrices:
finv(x) = (Dx-B)/(-Cx+A) =
Notice the similarity to the inverse of the 2x2 matrix (A B);(C D) is (D/(AD-BC) -B/(AD-BC));(-C/(AD-BC). A/(AD-BC))
I’m not the first person to notice this but I can’t find a satisfactory explanation for this that goes beyond “well they’re both inverses”. I’d really like to understand if there’s a deeper connection that goes beyond interesting coincidence. Otherwise, it’s just a nice mnemonic shortcut to finding the inverse of a linear/linear rational function.

[jensraab]

Cool!
I don't think I've ever seen this before.

[lucatherine]

f[3x/(2x +1)] = 4/(5x -6) find f(x):
Let y = 3x/(2x +1) ⇒ 2xy +y -3x = 0
⇒ x (2y -3) = -y ⇒ x = y/(3 -2y)
⇒ f(y) = 4/[5y/(3 -2y) -6] = 4 (3 -2y)/(17y -18)
⇒ f(x) = (12 -8x)/(17x -18)

[jacobgoldman]

What about the valid domain can’t x not be -0.5 or 1.2 in initial functional equation but it would be ok in the new one?

[tassiedevil]

Given that bprp used the term inverse function, perhaps it would have been conceptually clearer to spell out the composition of functions and say that f(g(x))=4/(5x-6), with g(x) =3x/(2x+1) as the function to be "inverted". So setting t=g(x), the search is for x=g_inverse(t) to substitute into the original RHS.

[flavioferrari-oss]

Boas festas e muita saúde professor! 👍🙏🙏🙏

[superkilleryt]

i was thinking i hard trying to understanf. Just to realisd is just composite function.

[antonyqueen]

Very straightforward.
But still, no need to make the complicated calculation in the second part of the solution which is prone to errors.
Since we said let t=3x/(2x+1)
then once x(t) is found,
We simply write:
f(t)=4/(5x(t)-6) and replace x(t) by its value in terms of t.

[waylluq]

Can you suggest a book where I can found a chapter dedicated to functional equations and exercises?

[turtleboi]

why can you replace t with x in the last step?

[GravityTale]

Damg, Ik this is simple but it still excites me.

[f-th]

There's a "trick" to this type of question. (there are reasons behind this and it's not a coincidence)

Any function of the form f(x) = (ax+b)/(cx+d) can be thought of as a 2x2 matrix like this:
[a, b]
[c, d]
To get the inverse function, you just invert this matrix. If D is the determinant, then the inverse matrix is:
[d/D, -b/D]
[-c/D, a/D]
So the inverse of f is (dx/D -b/D)/(-cx/D + a/D), which you can simplify further by multiplying the numerator and denominator by D to get (dx -b)/(-cx + a), which is much cleaner.
(note that the case when D = 0, which would prevent us from dividing by D, is really just the case when the top row is a multiple of the bottom row, and the entire function would simplify to a constant, so it never arises)

If you want to compose 2 such functions, you would multiply their matrices, then take that matrix and reinterpret it back as a function.
In the case of this question, you have g(x) = 3x/(2x+1) and h(x) = 4/(5x-6), and you're given that f(g(x)) = h(x). If we let k(x) be the inverse of g, then to get f(x), you need to substitute x with k(x) and get f(x) = h(k(x))

Apply the above to the question, and you get that the matrix for k is
[1, 0]
[-2, 3]
and the matrix for h is
[0, 4]
[5, -6]
and then you multiply them (in the correct order, which is h*k) to get
[-8, 12]
[17, -18]
and this is the function f(x) = (-8x+12)/(17x-18), just like the video.

[DavidMFChapman]

let y = 3x /(2x + 1)
y(2x +1) = 3x
x(3 – 2y) = y
x = y/(3 - 2y)
5x – 6 = 5y/(3 – 2y) – 64
= (5y – 6(3 – 2y))/(3 – 2y)
= (17y – 18)/(3 - 2y)
4/(5x – 6) = 4(3–2y)/(17y – 18)
f(x) = 4(3-2x)/(17x-18)

[YassJ-jdsz]

Hi. Why do you specify "for precalculus students"? Is there another way to solve it with calculus?

[Ouari-kn]

I would like some explanations If x=0 we have f(0) = F(0) and f(0)= F(0) = -4\6 For x=1 we have f(1)=F(1)= -4 ok but if x=2 we have f(6\5) and f(6\5) does not exist but F(6\5)=-4\11