The function f : [a, infinity) -> R with rule f (x) = 2x^3 − 3x^2 + 6 will have an inverse function provided?
the answer was a >= 1,
but why not also a <= 0?
Hello, scorpion,
as you know a function f has an inverse function if f is monotonously increasing or monotonously decreasing.
So your function has 3 intervalls where f is monotonous:
$\displaystyle (-\infty;0],\ (0;1], (1,+\infty)$
So you are looking for a lower bound of the intervall which contains +infinity.
With your suggestion you would look for the upper bound.
EB
I have to send you this post without checking my text because the preview doesn't work. So watch out: There are possibly some mistakes.
Earboth, please understand I'm not picking on you, but that sentence just had me rolling on the floor.
Monotonous: (definition) tediously uniform or unvarying.
The word in English you were looking for is "monotonically."
It very much reminds me of a lady at a school I once worked at. She was trying to make a poster thanking everyone who had contributed money to the school's fundraising campaign. She wrote "Thank you for patronizing us."
For those who don't know, there are two definitions for the word "patronize"
1) to act as patron of : provide aid or support for
2) to adopt an air of condescension toward : treat haughtily or coolly
Two VERY different meanings!
-Dan
I understand what earboth said, but formally there is no answer to your question!!
A function,
$\displaystyle f:[a,\infty)\to \mathbb{R}$ would have an inverse if and only if $\displaystyle f(x)=2x^3-3x^2+6, x\geq a$ is a bijective map. That means is is a surjective map. But for $\displaystyle a\geq 1$ the map is not sujective. Because if you chose $\displaystyle y=0$ there is no $\displaystyle x\geq 1$ such as, $\displaystyle f(x)=0$. What you should have said is,
$\displaystyle f:[a,\infty)\to [5,\infty)$