https://www.youtube.com/watch?time_continue=335&v=u9v_bakOIcU At 5:35 What he's saying makes no sense to me. How is it a different function if x can be equal to 8? Please explain it in simpler and step by step terms because I don't understand what he's talking about.
(f/g)(x) x cannot be equal to -8. Why?
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Dr.Peterson
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The most important idea is that a function is defined by two things: its domain, and what it does to each element in the domain. If two functions have different domains, they can't be called the same function, even if they do the same to every element they both apply to.
This function, f/g, is not defined for x=-8 because, as you say, that results in division by zero. After canceling, that no longer happens, so the domain has been changed -- in fact, the simplified function can be thought of as an extension of f/g, since it has the same value on the domain of f/g, but is also defined at x=-8, extending the domain. In order to make the new function the same as the original, we explicitly state the domain restriction, reducing it back to the original domain.
I am not disagreeing with Dr. Patterson. I am simply saying the same thing in different words.
\(\displaystyle f(x) = \dfrac{x^2 - 2x - 3}{x + 1} \text { for all } x \ne -\ 1.\)
\(\displaystyle g(x) = x - 3 \text { for all } x.\)
You can't say f(x) = g(x) or f(x) and g(x) are the same because g(-1) = - 4 and f(-1) does not equal anything at all.
Make sense?
To say two functions are equal or the same means that they have the same domains and give the same result for every element in the domain.
You could say in the above example that f(x) = g(x) for all positive x because the functions give the same result for that limited domain.
\(\displaystyle f(x) = \dfrac{x^2 - 2x - 3}{x + 1} \text { for all } x \ne -\ 1.\)
\(\displaystyle g(x) = x - 3 \text { for all } x.\)
You can't say f(x) = g(x) or f(x) and g(x) are the same because g(-1) = - 4 and f(-1) does not equal anything at all.
Make sense?
To say two functions are equal or the same means that they have the same domains and give the same result for every element in the domain.
You could say in the above example that f(x) = g(x) for all positive x because the functions give the same result for that limited domain.