0
votes
0answers
61 views

Calculus - Proof composition of two function.

$f,g$ are defined in $\mathbb R$ and these conditions apply. $$1) \lim_{x \to x_0} g(x) = y_0$$ $$2) \lim_{y \to y_0} f(y) = L (L\in\mathbb R)$$ $$3)\mbox{ There is a pocked enviroment of }x_0 \mbox{ ...
3
votes
2answers
47 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
2
votes
0answers
27 views

Rendering the derivative of composite functions from a graph

I'm on a workbook problem and I want to make sure I'm doing it properly. The problem asks me to find the derivatives of composite functions when given only the graphs of the original functions, here ...
0
votes
6answers
1k views

Application of Composition of Functions: Real world examples?

Do you know of a real world example where you'd combine two functions into a composite function? I see this topic in Algebra 2 textbooks, but rarely see actual applications of it. It's usually plug ...
1
vote
2answers
97 views

Maximal domain for composite functions.

Question $\mathbf 5$ If $f:(-\infty,1)\to R$, $f(x)=2\log_{\,e}(1-x)$ and $g:[-1,\infty)\to R$, $g(x)=3\sqrt{x+1}$, then the maximal domain of the function $f+g$ is ...
0
votes
2answers
92 views

function composition - n times

Please consider this function: $$f(x) = \frac{x}{{\sqrt[6]{{1 + {x^6}}}}} $$ What would be the value of the composition (n times): $$f \circ f... \circ f = ? $$ I tried doing it manually, maybe ...
0
votes
1answer
27 views

Finding a formula using two other functions.

I am trying to find a formula for $p(x)$, but I am not seeing where to start. I have two functions $m(x)=9*27^x$ and $n(x)=9^x$ I need to find the formula for function $p(x)$ if $m(x) = p(n(x))$ ...
0
votes
0answers
53 views

The limit of a composite function

I am presented with the following task: "You are given that $k(h) \neq 0$ and $h \neq 0$. If $\lim_{k \to 0} F(k) = L$ and $\lim_{h \to 0} k(h) = 0$, show that $\lim_{h \to 0}F(k(h)) = L$. This is ...
1
vote
3answers
307 views

Left Inverse: An Analysis on Injectivity

I'm told that $g$ is a left inverse of $f$ if $g\circ f=1_X$. I'm also told that if $f$ has a left inverse, then $f$ must be injective. I'm now asked to prove the converse, namely that if ...
0
votes
3answers
34 views

Derivatives of Functions

Suppose $ F(x)=f(g(x)) $ $g(1)=3$ $g'(1)=4$ $f'(1)=6$ $f'(3)=5$ What is $F'(1)$ ?
5
votes
4answers
303 views

3rd iterate of a continuous function equals identity function

If $ f: \mathbb{R} \to \mathbb{R} $ is continuous, and $\forall x \in \mathbb{R} :\;(f \circ f \circ f)(x) = x $, show that $ f(x) = x $. The condition that $f$ is continuous on $\mathbb{R}$ is ...