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.

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 […]

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 […]

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 […]

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.

(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 […]

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 […]

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 […]

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 […]

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

Intereting Posts

What is a constructive proof of $\lnot\lnot(P\vee\lnot P)$?
A variational strategy for finding a family of curves?
How to solve $ \sqrt{x^2 +\sqrt{4x^2 +\sqrt{16x^2+ \sqrt{64x^2+\dotsb} } } } =5\,$?
Finding a parabola from three points algebraically
I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
Book request: Mathematical Finance, Stochastic PDEs
Integral domain with fraction field equal to $\mathbb{R}$
How to fit $\sum{n^{2}x^{n}}$ into a generating function?
Fixed Points and Graphical Analysis
For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$
How to determine highest power of $2$ in $3^{1024}-1$?
If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?
Can every closed subspace be realized as kernel of a bounded linear operator from a Banach space to itself?
Compact Operator defined by inner product
Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt{\cos(2\pi/7)}+\sqrt{\cos(4\pi/7)}+\sqrt{\cos(8\pi/7)}$