Given the function f(x)=x^2-x^4, calculate (f o f o f o f)(1) .

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have f(x) = x^2 - x^4

(f o f o f o f)(1)

=> (f o f o f)(1^2 - 1^4)

=> (f o f o f)(0)

=> (f o f)(0^2 - 0^4)

=> (f o f)(0)

=> f(0)

=> 0

The required value of (f o f o f o f)(1) = 0

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll determine first (fofofof)(x)

(fofofof)(x)=f(f(f(f(x))))

fof(x)=f(f(x))=[f(x)]^2-[f(x)]^4

fof(x)=(x^2-x^4)^4=(x^2-x^4)^2 * [1-(x^2-x^4)^2]

fof(x)=(x^2-x^4)^2(1-x^2+x^4)(1+x^2-x^4)

Well put (fof)(x)=g(x)

f(f(f(x)))=f(g(x))=[g(x)]^2-[g(x)]^4

f(f(f(x)))=f(g(x))=(x^2-x^4)^4[1-(x^2-x^4)^2]^2-(x^2-x^4)^8[1-(x^2-x^4)^2]^4

h(x)=(x^2-x^4)[1-(x^2-x^4)^2]^2*{1-(x^2-x^4)^2[1-(x^2-x^4)^2]^2}

f(h(x))=(x^2-x^4)^8 *[1-(x^2-x^4)^2]^4 {1-(x^2-x^4)^2 [1-(x^2-x^4)^2]^2}^2-(x^2-x^4)^32*[1-(x^2-x^4)^2]^16*{1-(x^2-x^4)^2*[1-(x^2-x^4)^2]^2}^8

Now , we'll determine(fofofof)(1)=f(h(1))

f(h(1))=(1^2-1^4)^8*[1-(1^2-1^4)^2]^4*{1-(1^2-1^4)*[1-(1^2-1^4)^2]^2}^2-(1^2-1^4)^32*[1-(1^2-1^4)^2]^16*{1-(1^2-1^4)[1-(1^2-1^4)[1-(1^2-1^4)^2]^2}^8

f(h(1))=0*1(1-0)-0*(1-0)(1-0)=0

The value of composition of functions is (fofofof)(1)=0.

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