# Supremum and Infimum of functions

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?

• Do you know what the words monotone, injective, surjective, bijective, supremum, and infimum mean? If not, look up their definitions and if you still don't understand what some of them mean, feel free to ask a more specific question here. If you know what the words mean, but don't understand how to determine whether a function is monotone or not and so on, make sure to make that clear in your question. It is very hard to tackle the issues you're struggling with if you don't explicitly say what those issues are. – Michael Albanese Jul 17 '13 at 9:00
• The supremum of a function is the supremum of its image. A function $f:D \to \mathbb R$ is said to attain its supremum if there exists $x \in D$ with $f(x) = \sup f(D)$. Since the supremum of an arbitrary subset $I \subseteq \mathbb R$, if exists, may not be in $I$, it is possible for a function $f$ to not attain the supremum (even when the supremum exists). – Tunococ Jul 17 '13 at 22:14

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.