I have been given the following homework problem... struggling. Any help would be appreciated.
With the following functions state;
a) State if the function is monotone. 1
b) Decide if it is injective, surjective or bijective on the given domain.
c)Find the Supremum and Infimum (if they exist); in each case state whether or not the function attains its bounds.
$$1.\ \ f(x)=\frac1{1+x^4}:\mathbb{R}\to(0,1]$$ $$2.\ \ f(x)=\tan x:(-\frac\pi2,\frac\pi2)\to\mathbb{R}$$ I understand;
- Injective is one to one (does monotone check a similar thing?)
- Supremum is the greatestlower bound of the set (given the constraints, $\mathbb{R}$/$\mathbb{N}$/$\mathbb{Z}$/etc)
- Infimum is the lowest upper bound of the set (given the constraints, $\mathbb{R}$/$\mathbb{N}$/$\mathbb{Z}$/etc)
How do you apply these when you are given functions, rather than sets?
$\endgroup$ 21 Answer
$\begingroup$Pictures should significantly help with these questions. Look at the pictures in this link of your first function.
Now, a function is monotonic, roughly, if it always increases or always decreases. From the picture, we can see that $1/(1+x^4)$ is not monotonic.
Injective means that every $y$ value has one and only one $x$ value. Notice that the $y$ value $1/2$ corresponds to the $x$ values $1$ and $-1$, so the function is not injective. Another way to test this is to use the horizontal line test.
The function is surjective, because every $y$-value within $(0,1]$ is obtained.
The function cannot be bijective, because it has to be both injective and surjective, but it is not injective.
Finally, the supremum and infimum of a function are the least upper bound and greatest lower bound respectively. Think of them as the $y$-values corresponding to the horizontal lines that enclose the entire graph. In this case, it will be the horizontal lines $y=0$ and $y=1$, so the infimum is $0$ and the supremum is $1$. $y=1$ actually intersects the function, so the supremum is obtained. $y=0$ does not intersect the function, so the infimum is not obtained.
See if you can work with $\tan x$ on your own.
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