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f of f of x or f((x)) and its general term.
Equation of function of a function is given in terms of single variable then find the equation of that function.
Here I will show you the general case also.
Given equation:
\[f(f(x))=x^4-6x^3+6x^2+9x\]
Solution 1:
By converting given equation in the form of f(f(x)):
Given equation:
\[x^4-6x^3+6x^2+9x\]
\[\Rightarrow (x^2)^2-2(x^2)(3x)+\color{blue} {9x^2-9x^2}+6x^2+9x\]
\[\Rightarrow ((x^2)^2-2(x^2)(3x)+9x^2)-(3x^2-9x)\]
\[\Rightarrow (x^2-3x)^2-3(x^2-3x)\]
It is in the form of f(f(x)):
\[f(\color{fuchsia} {f(x)})= (\color{fuchsia}{x^2-3x})^2-3(\color{fuchsia}{x^2-3x})\]
\[\Rightarrow f(x)=x^2-3x\]
By this method we get:
f(x) = x² - 3x
Solution 2: (General solution)
By comparing coefficients with genera term:
Observe that,
If the highest power of f(x) is one;
Then highest power of f(f(x)) will be two;
Also highest power of f(f(f(x))) will be three:
And so on,
Similarly,
If the highest power of f(x) is two;
Then highest power of f(f(x)) will be four;
Also highest power of f(f(f(x))) will be six:
And so on,
If the highest power of f(x) is three;
Then highest power of f(f(x)) will be six;
Also highest power of f(f(f(x))) will be nine:
And so on,
This will be same for all kinds of functions,
By using this, we can create the general terms for given function.
Given equation:
\[x^4-6x^3+6x^2+9x\]
This equation is for f(f(x)) and its highest power is four.
⇒ The highest power of f(x) will be two.
⇒ f(x) = ax² + bx + c
Also, observe that the constant term is not given in the equation that means the constant in the equation is '0'.
⇒ Constant in f(x) will also be '0'.
⇒ f(x) = ax² + bx
By considering above equation, create f(f(x)):
\[\Rightarrow f(f(x))=a(ax^2+bx)^2+b(ax^2+bx)\]
\[\Rightarrow f(f(x))=a(a^2x^4+2abx^3+b^2x^2)+b(ax^2+bx)\]
\[\Rightarrow f(f(x))=a^3x^4+2a^2bx^3+ab^2x^2+abx^2+b^2x\]
\[\Rightarrow f(f(x))=a^3x^4+2a^2bx^3+(ab^2+ab)x^2+b^2x\]
Now compare this equation with the given equation:
\[x^4-6x^3+6x^2+9x=a^3x^4+2a^2bx^3+(ab^2+ab)x^2+b^2x\]
We get four equations by this,
\[a^3=1\]
\[2a^2b=-6\]
\[ab^2+ab=6\]
\[b^2=9\]
By solving these equations, we get:
\[(a,b)=(1,-3)\]
By this our equation will be:
\[f(x)=x^2-3x\]
By this method we get:
f(x) = x² - 3x
Question related to Algebra, Function.
