Approximating different functions

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
Mentor
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')##.

kent davidge
Mark44
Mentor
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.