Articles of functions

Show that if $f \circ g$ is surjective, then $f$ is surjective, and $g$, the function applied first, needs not to be.

Show that if $f \circ g$ is surjective, then $f$ is surjective, and $g$, the function applied first, needs not to be. (Note:$f \circ g=f(g(s))$, $f$ and $g$ are well defined) This statement originates from http://en.wikipedia.org/wiki/Surjective_function

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am unsure how to show this, can anyone point me in the right direction?

Composition of a piecewise and non-piecewise function

Say you have 2 functions, one of which being a piecewise function: $f(x)= x^2+2, x<1$ or $2x^2+2, x>=1$ And the other: $g(x)=x^4+1$ How would you find the $f[g(x))]$? I understand regular function composition, I just don’t understand what to do when piecewise functions are involved.

General method for composition of piecewise defined functions

There is a similarity in questions about composition of functions piecewise defined (see e.g. here, here and here). In these questions the goal is always the same: Given $f,g$ piecewise defined, compute $f \circ g$. (see also example below) The aim of this question is to express the mechanism behind these exercises and give a […]

Algebratically find domain of $y=\ln \frac{3x-1}{x+2}$

I had this question : $y=\ln \frac{3x-1}{x+2}$ and to find the domain I managed to find most of it, that is, I figured out $x\neq -2$ cause of the denominator. I also found $x>1/3$ by setting the expression inside the $\ln()$ to larger than $0$. However, this did not produce the last solution of $x<-2$. […]

How many functions $f:S \rightarrow S$ satisfy $f(f(x)) = f(x)$ for all $x \in S$?

Let $S = \left\{ {1, 2, 3, 4, 5}\right\}$. How many functions $f:S \rightarrow S$ satisfy $f(f(x)) = f(x)$ for all $x \in S$?

Inverse function theorem question – multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) Show that $f$ is injective around every point in $\mathbb R^2$ – I managed to solve this 2) Find environments around the points $(0,\pi)$ and $(-1,\frac{\pi}{2})$ such that $f$ is […]

How many solutions does $\cos(97x)=x$ have?

How many solutions does $\cos(97x)=x$ have? I have plot the function. However I don’t know how to solve the problem without computer. Can anyone give a fast solution without a computer?

Necessary and Sufficient Conditions for Random Variables

I’ve been reading 2 textbooks in parallel on Probability Theory and they have 2 separate definitions of random variables $$ f:(\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{B}) \iff \forall B \in \mathcal{B}, \quad f^{-1}(B) \in \mathcal{S} $$ and $$ f:(\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{B}) \iff \forall x \in \mathbb{R}, \quad f^{-1}((-\infty,x]) \in \mathcal{S} $$ The text giving […]

Concavity of a function

Let g be a concave function such that $g[x] = (1+x) \log(1+x) – x\log(x)$, where $x>0$. Show that the following function is concave for given constraints on the parameters: $f[x] = g[\frac{1}{2}(\sqrt{(1+(1+a) x+(1-a) y)^2-4 a x (1+x)}+(1-a)x-(1-a)y-1)]+g[\frac{1}{2}(\sqrt{(1+(1+a) x+(1-a) y)^2-4 a x (1+x)}-(1-a)x+(1-a)y-1)] – g[(1- \frac{a}{(1-a)y+1})x],$ where $x>0, y\geq 0 $ and $0.5 \leq a \leq […]