This happens all the time: ##p=0## for ##f(x)=x^2\; , \;g(x)=\sin(x)\; , \;h(x)=|x|## etc. However ##f=g## if ##f(p)=g(p)## for all points ##p \in \operatorname{dom}(f)=\operatorname{dom}(g)##. We also widely use the fact, that functions are the same at one point and close in their neighborhood, when we approximate a function by its tangent in ##p##.kent davidge said:Today, while reading about bijections, a question came into my mind. And that is: is there any way that two different functions ##f## and ##g## acting on a same point ##p## gives the same output? In symbols, as I'm not good in English, is it possible that ##f (p) = g(p)## with ##f \neq g##?
Here's an example that's almost what you're talking about:kent davidge said:Today, while reading about bijections, a question came into my mind. And that is: is there any way that two different functions ##f## and ##g## acting on a same point ##p## gives the same output? In symbols, as I'm not good in English, is it possible that ##f (p) = g(p)## with ##f \neq g##?
Equality between functions refers to the idea that two functions are considered equal if they produce the same output for every input. In other words, if two functions have the same domain and their outputs are equal for every input value, then they are considered equal.
To determine if two functions are equal, we can use the substitution method. This involves plugging in the same input values into both functions and comparing their outputs. If the outputs are the same for every input value, then the functions are considered equal.
Yes, there are certain cases where two functions may have the same output for every input value, but are not considered equal. This can occur if the two functions have different domains, or if they are defined differently but happen to produce the same output. In these cases, the functions are not considered equal.
Equality between functions is important in mathematics because it allows us to compare and manipulate different functions. By understanding when two functions are equal, we can simplify equations and solve problems more efficiently.
Function composition involves combining two functions to create a new function. In order for function composition to be valid, the two functions must be considered equal. This means that their outputs must be the same for every input value. So, equality between functions is crucial in determining the validity of function composition.