Solving A Functional System of Equations in Two Ways

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[vencedore]

I just found the inverse of g(x) and used it on the first equation.
Just realized it's literally the second solution lol

[ozandeste]

*4th* method:
f(x) = ax^2 + bx + c

To find the constant of f function g(x) = 0 then you will find f(0) = 43. Then, we need to use f(1) and f(-1) to find a and b. a + b = 14, a + b = -12; a = 1, b = 13, c = 43.

[YossiSirote]

I used a third method. I let f(y)=ay^2+by+c, then plugged in f(g(x)) and solved for a, b, and c, by matching x^2’s, x’s, and constant terms.

[rsri]

Since f(2x-7) = 4x^2-2.x+1

=> f(x) is of form ax^2+b.x+c

=>f(2x-7) = a.(2x-7)^2 +b.(2x-7) +c

Some algebra and comparing coefficients

4a=4 => a=1, 28a-2b = 2 => b=13, 49a-7b+c =1 => c=43.

f(x) = x^2 + 13x + 43

[WahranRai]

For method 1 . Division of polynomials or synthetic division gives :
4x^2 - 2x + 1 = (2x -7)(2x+6) + 43 = (2x-7)(2x -7 + 13) + 43 = (2x-7)^2 + 13(2x-7) + 43
then 4x^2 - 2x + 1 = f(x)^2 + 13f(x) + 43 etc...

[aduedc]

You made typo mistake on your 1st method results, the correct answer is F(X)=X^2+13X+43 and not F(X)=X^2+13g(x)+43

[JohnRandomness]

Okay g(x) = 2x - 7 is invertible: x(g) = (g + 7)/2. Therefore, f(g) = 4((g + 7)/2)² - 2((g + 7)/2) + 1 = g² + 13g + 43.

Therefore, f(x) = x² + 13x + 43.

[arinelakshmisaikumar]

I have solved this problem using first method but after seeing video I realized that second method is quite simple . Thanks sir for this problem

[raystinger]

So the 4x² term in f(g(x)), given that g(x) has a 2x term in it, makes it reasonable to assume that f(x) is a quadratic polynomial of the form:
f(x) = (x - p)² + q
From that, applying x=g(x)=2x + 7, we get, in the end:
p=-13/2 and q=1/4
Expanding the final equation, we get:
f(x) = x² + 13x + 43

[waiphyoemg]

In our country we usually use the second method when we r given to solve such kind of problem. Whatever it takes, love both methods 😍

[ahmadmazbouh]

I used the first method this time
I appreciate showing more than one method cuz it helps me looking at the problem from other prspectives

[whuvdzw]

f(x)=ax²+bx+c
fₒg(x)=a(g(x))²+b(g(x))+c
the equation given to us;fₒg(x)=4x²-2x+1
they are identical equations.
...
this is how I solved it.
but I realized it is not better...

[TheFinav]

Error at 5'38". 13x not 13g(x)

[armacham]

Here is a fourth method:

F(2x - 7) = 4x^2 – 2x + 1

take the derivative of both sides. Use the chain rule of course. Where the derivative of 2x-7 is equal to 2

f'(2x-7)*(2) = 8x - 2

f'(2x-7) = 4x - 1

a = 2x - 7

f'(a) = 4x - 1 - 13 + 13

f'(a) = 2a + 13

Integrate both sides with respect to a

f(a) = a^2 + 13a + c

from there it is trivial to solve for the value of c

[henrymarkson]

The second method recognises that f(2x-7) is a linear transformation of f(x)
To retrieve f(x), we reverse the transformation: 2x-7=x' thus x=(x'+7)/2.
Substitute x=(x'+7)/2 into 4x^2-2x+1

[mikezilberbrand]

Instead of using y, just find x in terms of g. x= (g+7)/2. Then f(x) = f((g+7)/2)= 4((g+7)/2)^2-2(g+7)/2+1= (g+7)^2 -(g+7)+1= g^2+14g+49-g-7+1=g^2+13g+43, then replace g with x. f(x)= x^2+13x+43.

[communist_squad]

There is a more simple method based on the first one. As we know, the derivative of the complex function f(g(x)) will be the following:

(f(g(x)))'=f'(g(x))*g'(x)

This equation is true for any f() and g(). So we take derivatives for f(g(x)) and for g(x). They will be:

(f(g(x)))'=8x-2;
g'(x)=2.

And according to the equation above, we find f'(g(x)) through dividing the (f(g(x)))' by g'(x). It will be:

f'(g(x))=4x-1.

And now we put 2x-7=t. So 2t=4x-14. And cause f'(g(x))=4x-1=(4x-14)+13, f'(t) will be:

f'(t)=2t+13.

And now we take an integral to find f(t), which is an antiderivative of f'(t). It will be:

f(t)=t²+13t+C,

...where C is an unknown constant, which is to be found. To find C we take the first formula of f(g(x)) and equate it to the formula above, while replacing t with (2x-7). So we get:

4x²-2x+1=(2x-7)²+13(2x-7)+C;
4x²-2x+1=4х²-28х+49+26х-91+C;
-2x+1=-2x-42+C;
1=C-42;
C=43.

And after C is successfully found, we get the accurate formula of f(t):

f(t)=t²+13t+43.

And by replacing t with x, we finally get f(x):

f(x)=x²+13x+43.

[swenji]

This is not sufficient to conclude unless you prove that g is surjective. Finding the inverse of g does everything you need

[kassuskassus]

g(x)=2x-7, so 2x=g(x)+7, then 4x^2 =g(x)^2+14g(x)+49. We replace all this in the expression of f(g(x)) and substitute g(x) by x.

[GlorifiedTruth]

Another great video. Videos like this one are fun and help me get over my distaste for composite functions.