An algebra problem by I Gede Arya Raditya Parameswara

Algebra Level 5

Suppose that f : R → R f: \mathbb{R} \rightarrow \mathbb{R} satisfies

f ( 6 f ( a ) − 36 ) − f ( 6 b ) a − b = 6 , for a ≠ b \frac{f\big(6f(a)-36\big)-f(6b)}{a-b}=6, \quad \text{ for } a\neq b

Given the above, what is the value of f ( 6 ) ? f(6)?


The answer is 12.

I Gede Arya Raditya Parameswara by I Gede Arya Raditya Para… , 17

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2 solutions

Notice that the second one, f ( 6 b ) f(6b) , can take the same value as f ( 6 f ( a ) − 36 ) f(6f(a)-36) by substituting b = f ( a ) − 6 b=f(a)-6 . Therefore, if we suppose that f ( a ) − 6 ≠ a f(a)-6 \not= a (which comes from the supposition that b ≠ a b \not= a ) for some real number a a , we would get f ( 6 f ( a ) − 36 ) − f ( 6 f ( a ) − 36 ) a − f ( a ) + 6 = 6. \frac{f(6f(a)-36)-f(6f(a)-36)}{a-f(a)+6}=6. However, the enumerator of the fraction on the left hand side is zero, which makes this equation fails. Therefore, f ( a ) − 6 = a f(a)-6=a for all real number a a . Consequently, f ( 6 ) = 12 f(6)=12 .

3 Helpful 1 Interesting 0 Brilliant 0 Confused

f ( x ) f(x) = x + 6 x+6 .

It can be achieved by taking f ( x ) f(x) = x + c x+c , c is any real no. (For proof see note)

N o t e : Note: Or partial differentiation w.r.t 'b' by sending (a-b) to R.H.S,

We get f ′ ( 6 b ) f'(6b) = 1 1 ,

So , it should be a linear function.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Is that cauchy?

I Gede Arya Raditya Parameswara - 4 years ago

How do you know that the function is differentiable? That is currently an assumption and has to be proven.

Calvin Lin Staff - 4 years ago

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