1. ## questions on functions

Given that
f(x)= $\displaystyle \frac{5-x}{1-x}$

(a) Explain why f has an inverse and show that $\displaystyle f^{-1}=f$.
(b) Evaluate $\displaystyle f^{51}(4)$.

i can do part a,
but have no idea how to start for part b.
any help is appreciated. =)

2. ## Re: questions on functions

I guess you've to calculate the 51th derivative of the function $\displaystyle f$, I should try to recognize a pattern in you derivatives.

3. ## Re: questions on functions

$\displaystyle f(f(x))= \frac{5- \frac{5- x}{1- x}}{1- \frac{5- x}{1- x}}$
Multiply both numerator and denominator by 1- x:
$\displaystyle f(f(x))= \frac{5(1- x)- (5- x)}{1- x- (5- x)}= \frac{ 5-5x-5+ x}{1- x-5+ x}= \frac{-4x}{-4}= x$

That shows both that f is invertible and that $\displaystyle f^{-1}(x)= f(x)$.

4. ## Re: questions on functions

Originally Posted by wintersoltice
Given that f(x)= $\displaystyle \frac{5-x}{1-x}$
(a) Explain why f has an inverse and show that $\displaystyle f^{-1}=f$.

(b) Evaluate $\displaystyle f^{51}(4)$.
I think that the notation $\displaystyle f^{51}$ refers to function composition.
For example: $\displaystyle f^5 (x) = f \circ f \circ f \circ f \circ f(x) = f\left( {f\left( {f\left( {f\left( {f(x)} \right)} \right)} \right)} \right) = ?$

5. ## Re: questions on functions

Good point! And the fact that f is its own inverse makes that trivial!

6. ## Re: questions on functions

i found the solution from my school's online portal.

but i don't understand the steps. i mean from step 2 to 3.

step1$\displaystyle f^{51} (4)$
step2$\displaystyle =f[f^{50}(4)]$
step3$\displaystyle =f(4)$ as $\displaystyle (# f^2(x)=f^{-1}f(x)=x)$
step4$\displaystyle =\frac{5-4}{1-4}$
step5$\displaystyle =\frac{-1}{3}$

7. ## Re: questions on functions

Because $\displaystyle f=f^{-1}$ that means $\displaystyle f^2(x)=f\circ f(x)=f(f(x))=x$.
So $\displaystyle f^5(x)=f(f^4(x))=f(x)$.

8. ## Re: questions on functions

Originally Posted by Siron
I guess you've to calculate the 51th derivative of the function $\displaystyle f$, I should try to recognize a pattern in you derivatives.
That would be true if the notation were, instead,

$\displaystyle f^{(51)}(x),$

with the parentheses in the exponent.