Articles of function composition

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 […]

The limit of composition of two functions

I need your help in solving this limits problem. Let $f$ and $g$ be two functions defined everywhere. If $\lim_{u\to b} f(u) = c$ and $\lim_{x\to a} g(x) = b$, then you may believe that $\lim_{x\to a} f(g(x)) = c$. This problem shows that this is not always true. Consider functions $f$ and $g$ defined […]

Why does Arccos(Sin(x)) look like this??

I can kind of understand the main direction (slope) of $y$ over the different $x$ intervals, but I can’t figure out why the values of $y$ take on the shape of straight lines and not curves looking more like those of sin, cos… EDIT: I understand that the derivative of Arccos(Sin(x)) gives 1 or -1 […]

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.

Can the composition of two non-invertible functions be invertible?

(Context: I came across this exercise in the textbook “Coding the Matrix” when reading it to supplement my studies in the Coursera class “Coding the Matrix”.) After proving that the composition of invertible functions is itself invertible (by showing that a function is invertible iff it is both one-to-one and onto), the author left as […]

Operational details (Implementation) of Kneser's method of fractional iteration of function $\exp(x)$?

For a long time (a couple of years) I’m following the Q&A’s about “half-iterate of $\exp(x)$” etc. where there exists a $\mathbb C \to \mathbb C$ due to Schröder’s method, but also a $\mathbb R \to \mathbb R$ for fractional heights $h$ due to Hellmuth Kneser. I would like to understand the latter method of […]

Every two positive integers are related by a composition of these two functions?

How would one prove/disprove this? … Conjecture: Suppose $p$, $q$ are distinct primes, and define $\ f(n) = n p, \ g(n) = \left \lfloor \frac{n}{q} \right \rfloor$ for all $n \in \mathbb{N_+}$; then for all $x,y \in \mathbb{N_+}$, there exists a composition $F = g \circ g \circ \cdots \circ g \circ f \circ […]

Proving that if two linear transformations are one-to-one and onto, then their composition is also.

I am attempting to solve a problem with the following given conditions: Let V, W. and Z be vector spaces, and let $T:V \longrightarrow W$ and $U: W\longrightarrow Z$ be linear.Prove that if U and T are one-to-one and onto, then UT is also. Here is my attempt for the one-to-one part: “Suppose that U […]

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{\left(\frac{\pi}{\alpha^5}\right)}^\frac{1}{2}$

This question already has an answer here: $ \int_{-\infty}^{\infty} x^4 e^{-ax^2} dx$ 4 answers