lemon14
Feb 11, 2012, 03:26 AM
I'm not sure how to deal with this type of exercises. Here is what I solved so far:
1.
f(x)=f(x^2), x \geq 0
f(x)=f(x^2)=f(x^4)=...=f(x^{2n})
f(x)=f(x^{2n})\Rightarrow f(x)= \lim_{n \to \infty} f(x^{2n})= f(\lim_{x \to \infty} x^{2n})
f(x)=0 if x=0 and f(x)= \infty if x>0
Is this possible? I don't think it's correct.
2.
f(2^x)=f(3^x), x \epsilon R
3^x = t \Rightarrow \log_3 {t} = x
f(2^{\log_3{t}})=f(t) \Rightarrow f(x)=f(2^{\log_3{x}})
What should I do next?
1.
f(x)=f(x^2), x \geq 0
f(x)=f(x^2)=f(x^4)=...=f(x^{2n})
f(x)=f(x^{2n})\Rightarrow f(x)= \lim_{n \to \infty} f(x^{2n})= f(\lim_{x \to \infty} x^{2n})
f(x)=0 if x=0 and f(x)= \infty if x>0
Is this possible? I don't think it's correct.
2.
f(2^x)=f(3^x), x \epsilon R
3^x = t \Rightarrow \log_3 {t} = x
f(2^{\log_3{t}})=f(t) \Rightarrow f(x)=f(2^{\log_3{x}})
What should I do next?