Journey Stout
2022-01-22
Answered

What is the continuity of the composite function $f\left(g\left(x\right)\right)$ given $f\left(x\right)=\frac{1}{\sqrt{x}}$ and $g\left(x\right)=x-1$

You can still ask an expert for help

fumanchuecf

Answered 2022-01-23
Author has **21** answers

Explanation:

The process here is to verify first the domain of g(x) (the inner function in the composition).

The domain is$\{x\mid x>1,x\in \mathbb{R}\}$

$f\left(g\left(x\right)\right)=\frac{1}{\sqrt{x}}$

There will be two types of restriction on the domain in this problem.

- When the number underneath the$\sqrt{}$ is less than 0.

- When the denominator equals 0.

The number under the square root will be negative whenever$x<1$ . The denominator will equal 0 when $x=1$ , so the domain of the composition is

$\{x\mid x>1,x\in \mathbb{R}\}$

The process here is to verify first the domain of g(x) (the inner function in the composition).

The domain is

There will be two types of restriction on the domain in this problem.

- When the number underneath the

- When the denominator equals 0.

The number under the square root will be negative whenever

asked 2022-04-08

Find the value of $\mathrm{tan}}^{2}\frac{\pi}{16}+{\mathrm{tan}}^{2}\frac{2\pi}{16}+{\mathrm{tan}}^{2}\frac{3\pi}{16}+{\mathrm{tan}}^{2}\frac{4\pi}{16}+{\mathrm{tan}}^{2}\frac{5\pi}{16}+{\mathrm{tan}}^{2}\frac{6\pi}{16}+{\mathrm{tan}}^{2}\frac{7\pi}{16$

asked 2022-05-27

Number of roots of ${\mathrm{\Sigma}}_{r=1}^{2009}\frac{r}{x-r}$

asked 2022-06-25

Given that $({a}_{n}{)}_{n=1}^{\mathrm{\infty}}$ is a sequence for which ${a}_{n}/\mathrm{log}(n)\to 1$ as $n\to \mathrm{\infty}$, show that

$\underset{x\to 1}{lim}\frac{\sum _{n=0}^{\mathrm{\infty}}{a}_{n}{x}^{n\text{}}}{\frac{1}{1-x}\mathrm{log}\left(\frac{1}{1-x}\right)}=1$

$\underset{x\to 1}{lim}\frac{\sum _{n=0}^{\mathrm{\infty}}{a}_{n}{x}^{n\text{}}}{\frac{1}{1-x}\mathrm{log}\left(\frac{1}{1-x}\right)}=1$

asked 2021-02-05

Suppose that u,vu,v and w are vectors such that $\u27e8u,v\u27e9=6,\u27e8v,w\u27e9=-7,\u27e8u,w\u27e9=13\mid \mid u\mid \mid =1,\mid \mid v\mid \mid =6,\mid \mid w\mid \mid =19\mid \mid u\mid \mid =1$, given expression $\u27e8u+v,u+w\u27e9$.

asked 2021-12-16

f(x)=105√x3−√x7+63√x8−3f(x)=10x35−x7+6x83−3

asked 2021-11-16

Approximate f by a Taylor polynomial with degree n at the number a.

$f\left(x\right)=\mathrm{ln}(1+2x),\text{}a=1,\text{}n=3,0.5\le x\le 1.5$

asked 2022-07-14

Let $e(n)$ be the number of partitions of $n$ with even number of even parts and let $o(n)$ denote the number of partitions with odd number of even parts. In Enumerative Combinatorics 1, it is claimed that it is easy to see that $\sum _{n\ge 0}(e(n)-o(n)){x}^{n}=\frac{1}{(1-x)\times (1+{x}^{2})\times (1-{x}^{3})\times (1+{x}^{4})\times ...}$. I have been racking my head over this for the past few hours, and I can't see any light.

Noticed, that $e(n)-o(n)=2e(n)-p(n)$ where $p(n)$ is the number of partitions of $n$, so the above claim is equivalent to showing $\sum _{n\ge 0}e(n){x}^{n}=\frac{1}{2}\frac{1}{(1-x)(1-{x}^{3})(1-{x}^{5})...}(\frac{1}{(1-{x}^{2})(1-{x}^{4})....}+\frac{1}{(1+{x}^{2})(1+{x}^{4})........})$, and similarly, it is equivalent to $\sum _{n\ge 0}o(n){x}^{n}=\frac{1}{2}\frac{1}{(1-x)(1-{x}^{3})(1-{x}^{5})...}(\frac{1}{(1-{x}^{2})(1-{x}^{4})....}-\frac{1}{(1+{x}^{2})(1+{x}^{4})........})$, but these identities appear more difficult than the original one.

Noticed, that $e(n)-o(n)=2e(n)-p(n)$ where $p(n)$ is the number of partitions of $n$, so the above claim is equivalent to showing $\sum _{n\ge 0}e(n){x}^{n}=\frac{1}{2}\frac{1}{(1-x)(1-{x}^{3})(1-{x}^{5})...}(\frac{1}{(1-{x}^{2})(1-{x}^{4})....}+\frac{1}{(1+{x}^{2})(1+{x}^{4})........})$, and similarly, it is equivalent to $\sum _{n\ge 0}o(n){x}^{n}=\frac{1}{2}\frac{1}{(1-x)(1-{x}^{3})(1-{x}^{5})...}(\frac{1}{(1-{x}^{2})(1-{x}^{4})....}-\frac{1}{(1+{x}^{2})(1+{x}^{4})........})$, but these identities appear more difficult than the original one.