Approximating different functions

[kent davidge]

I have a question regarding different functions. Suppose we have two functions $f$ and $f'$ with same domain, but different codomains.

Consider that $f': x' \mapsto f'(x')$ and $f: x \mapsto f(x)$. If $x' = x + \sigma$, with $|\sigma| << 1$, can we say that $f'(x') \approx f(x')$?

 

[fresh_42]

No. Not in this generality. Let's say $f(x)=x^{-2}\; , \;f\,'(x)=x^{-1}$. Then we have $x'=-{10}^{-n}=x+\sigma = {10}^{-n}-2\cdot {10}^{-n}$ and $|\sigma|=|2\cdot {10}^{-n}| << 1$ but $-{10}^{n} =f\,'(x') \not \approx {10}^{2n}=f(x')$.

 

[Mark44]

I have a question regarding different functions. Suppose we have two functions $f$ and $f'$ with same domain, but different codomains.

Consider that $f': x' \mapsto f'(x')$ and $f: x \mapsto f(x)$. If $x' = x + \sigma$, with $|\sigma| << 1$, can we say that $f'(x') \approx f(x')$?

If you have two different functions, why not call them f and g? Or $f_1$ and $f_2$? Your notation $f'(x')$ looks like we're evaluating the derivative of f at a number x'.

 

[kent davidge]

If you have two different functions, why not call them f and g? Or $f_1$ and $f_2$? Your notation $f'(x')$ looks like we're evaluating the derivative of f at a number x'.

Yea, I agree. Coincidentally I was going to write them down as $f$ and $g$ in the opening post, but I used the more abstract notation.