prove that f(f(x)) is increasing if f(x) is a linear function.

Expert Answers
hala718 eNotes educator| Certified Educator

Since f is a linear fundtion, then:

Let f(x) = ax+b

f(f(x) = f(ax+b)

          = a(ax+b) +b

          = (a^2)x + ab + b

To find out if the function is increasing we need to determine 1st derivative:

==> f'(f(x) = a^2

a^2 is always positive value,  Then f'(fx) is an increasing function.

         

giorgiana1976 | Student

If f(x) is a linear function, we'll write it as:

 f(x)=ax+b

(fof)(x) = f(f(x)) = f(ax+b) = a(ax+b)+b

We'll remove the brackets:

(fof)(x) = a^2*x + a*b + b

In order to establish if f of f of x is an increasing function we have to calculate the first derivative of the (fof)(x).

[(fof)(x)]'= (a^2*x+a*b+b)'=a^2*1 + 0 + 0 = a^2

But a^2>0, for any value of a

If a^2>0, (fof)(x) > 0 

So, (fof)(x) is an increasing function!

neela | Student

 

Let  f(x) = ax+b  a linear function.

f(fx) = a (ax+b) +b = a^2x+b

d/dxf(f(x)) = (a^2x+b)' = a^2 which is positive.

So f(f(x)) is an increasing function.

 

thewriter | Student

let h(x)=f(f(x)).

f(f(x)) is increasing if the differential of f(f(x)) or h'(x) is positive. As f(x) is a linear function let it be equal to ax+b. Or f(x)=ax+b

So f(f(x))=f(ax+b)= a(ax+b)+b=a^2x+ab+b.

The differential of f(f(x)) is a^2 .

As a^2 is always positive, f(f(x)) is always an increasing function.

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