 # Practice Exercises for Composite Functions

Recent questions in Composite functions LoomiTymnk63x 2023-03-29

## Whether f is a function from Z to R if a) $f\left(n\right)=±n$. b) $f\left(n\right)=\sqrt{{n}^{2}+1}$. c) $f\left(n\right)=\frac{1}{{n}^{2}-4}$.? Preston Walker 2023-02-13

## Is 93 a prime or composite number? A)Prime number; B)Composite number; C)Cannot be determined Mara Boyd 2023-01-05

## If f(x)=x^2−1/x and g(x)=x+2/x−3, then the domain of f(x)/g(x) is... Mark Rosales 2022-11-18

## How to find domain of complicated composite functions. Find the domain of arccos(${e}^{x}$), are there universal steps I can take to be able to find the domain?

PrecalculusOpen question 2022-11-09

## write an equation for a rational function with:Vertical Asymptotes at x=2 and x=-6x-intercepts at x=-4 and x=-3y-intercept at 3 Martin Hart 2022-10-31

## Question about calculating the gradient of a composite function$\mathbf{r}=\left(x,y\right)=x\mathbf{i}+y\mathbf{j}$$‖r‖=\sqrt{{x}^{2}+{y}^{2}}$Let's assume that $\mathbf{r}\ne 0$We now define $f\left(x,y\right)={r}^{m}$What is the right expression for $\mathrm{\nabla }f$?1. $m{r}^{m-1}\mathbf{r}$2. $m{r}^{m-2}\mathbf{r}$3. $m{r}^{m-1}$4. $m{r}^{0.5m-1}\mathbf{r}$Reasoning was that it should be number $1$:the derivative of $f$ according to $x$ comes out as $m{r}^{m-1}\left(\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}\right)$, while the derivative of $f$ according to y comes out as $m{r}^{m-1}\left(\frac{y}{\sqrt{{x}^{2}+{y}^{2}}}\right)$. $\sqrt{{x}^{2}+{y}^{2}}=‖r‖=1$, and therefore this is equal to $m{r}^{m-1}\mathbf{r}$. adarascarlet80 2022-10-05

## Suppose that we have three $\mathbb{Z}\to \mathbb{Z}$ functions such as $f$, $g$ and $h$. How should $f$ and $h$ be so that f∘g∘h can be onto (surjective) given that $g$ is a one to one (injective) function? trapskrumcu 2022-09-26

## Chain rule for the derivative of a composite function$y=\left(\mathrm{sin}x{\right)}^{\sqrt{x}}.$ gaby131o 2022-09-26

## Show that if $\underset{x\to a}{lim}f\left(x\right)=L$, then $\underset{x\to a}{lim}cos\left(f\left(x\right)\right)=cos\left(L\right)$. videosfapaturqz 2022-09-24

## Let $f:\mathbb{D}\to \mathbb{D}$ (unit disk) be a holomorphic function with $f\left(0\right)=0,|{f}^{\prime }\left(0\right)|<1$. For ${f}_{n}=f\circ \cdots \circ f$, show that $\sum _{n=1}^{\mathrm{\infty }}{f}_{n}\left(z\right)$ converges uniformly on compact subsets in $\mathbb{D}$.I tried Schwarz lemma so that $|f\left(z\right)|\le |z|$, and I tried to use Weierstrass $M$ test, but I don't know how ${f}_{n}$ is bounded. How to solve this problem? malaana5k 2022-09-24

## Lets have $y:\mathbb{R}\to {\mathbb{R}}^{2}$ and that $f:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$, and lets assume that $f\left(y\left(x\right)\right)$ is given and that $y\left(x\right)=y\left({x}_{0}\right)+{\int }_{{x}_{0}}^{x}f\left(y\left(t\right)\right)dt$I'm a bit confused how there can be a function of $y\left(t\right)$ inside of the function definition for $y\left(x\right)$.I took the example that $y\left(x\right)=\left({x}^{2},x\right)$ and $f\left(y,z\right)=\left(y+z,y-z\right)$$⇒f\left(y\left(x\right)\right)=f\left({x}^{2},x\right)=\left({x}^{2}+x,{x}^{2}-x\right)$And now if we follow the definition of $y\left(x\right)$ we get:$y\left(x\right)=y\left({x}_{0}\right)+{\int }_{{x}_{0}}^{x}f\left(y\left(t\right)\right)dt$$y\left(x\right)=y\left({x}_{0}\right)+{\int }_{{x}_{0}}^{x}\left({t}^{2}+t,{t}^{2}-t\right)dt$$⇒y\left(x\right)=y\left({x}_{0}\right)+\left(\frac{1}{3}{t}^{3}+\frac{1}{2}{t}^{2},\frac{1}{3}{t}^{3}-\frac{1}{2}{t}^{2}\right){|}_{{x}_{0}}^{x}$$⇒y\left(x\right)=\left(\frac{1}{3}{x}^{3}+\frac{1}{2}{x}^{2}+{C}_{1},\frac{1}{3}{x}^{3}-\frac{1}{2}{x}^{2}+{C}_{2}\right)$Where is mistake? Camila Brandt 2022-09-23

## Find the indefinite integral of $x×\left(5x-1{\right)}^{19}$ by substitutionMy try:$u=5x-1$, so $\frac{du}{dx}=5$, thus $dx=\frac{du}{5}$How to cancel out the $x$ in front? unjulpild9b 2022-09-20

## Chain rule of partial derivatives for composite functions.Function of the form$f\left({x}^{2}+{y}^{2}\right)$How do I find the partial derivatives$\frac{\mathrm{\partial }f}{\mathrm{\partial }y},\frac{\mathrm{\partial }f}{\mathrm{\partial }x}$How $f\left({x}^{2}+{y}^{2}\right)$ behaves. Assuming it should of the form$g\left(x,y\right)\cdot 2y\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}h\left(x,y\right)\cdot 2x$ Liam Potter 2022-09-20

## Two continuous functions $f\left(x\right)$ and $g\left(x\right)$, is it possible that I expand $f\left(g\left(x\right)\right)$ at $g\left(0\right)$ using a series of $g\left(x\right)$?For example,$f\left(g\left(x\right)\right)=f\left(g\left(0\right)\right)+{f}^{\prime }\left(g\left(0\right)\right)g\left(x\right)+\frac{{f}^{″}\left(g\left(0\right)\right)}{2}{g}^{2}\left(x\right)+\cdots$In my case, $g\left(x\right)={e}^{-{x}^{2}}\left(0\le x\le 1\right)$. Aidyn Meza 2022-09-20

## Trying to calculate the value of $\frac{{\pi }^{4}}{90}$. Although I know the exact value (which I found on google to be $\frac{{\pi }^{4}}{90}$) but I wanted to derive it by myself. While doing so, I arrived at this rather peculiar expression: $C=\frac{7{ℼ}^{4}}{720}-\frac{1}{2}-\frac{P}{2}$where $C$ is the value of the composite zeta function at $2$ and $P$ is the prime zeta function at $2$. My question is this. What will be the value of $C$? vballa15ei 2022-09-14

## The question is:$f\left(x\right)=\frac{x}{x-1}$$g\left(x\right)=\frac{1}{x}$$h\left(x\right)={x}^{2}-1$Find $f\circ g\circ h$ and state its domain.The answer the textbook states is that the domain is all real values of $x$, except $±1$ and $±\sqrt{2}$.However surely the domain excludes $0$ as well, since $g\left(0\right)$ is undefined. Modelfino0g 2022-09-14

## Thorem: If $f\left(x\right)$ is continuous at $L$ and $\underset{x\to a}{lim}g\left(x\right)=L$, then $\underset{x\to a}{lim}f\left(g\left(x\right)=f\left(\underset{x\to a}{lim}g\left(x\right)\right)=f\left(L\right)$.Proof: Assume $f\left(x\right)$ is continuous at a point $L$, and that $\underset{x\to a}{lim}g\left(x\right)=L$.$\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta >0:\left[|x-L|<\delta \phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}|f\left(x\right)-f\left(L\right)|<\epsilon \right]$.And $\mathrm{\forall }\delta >0,\mathrm{\exists }{\delta }^{\prime }>0:\left[|x-a|<{\delta }^{\prime }\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}|g\left(x\right)-L<\delta \right]$.So, $\mathrm{\forall }\delta >0,\mathrm{\exists }{\delta }^{\prime }>0:\left[|x-a|<{\delta }^{\prime }\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}|f\left(g\left(x\right)\right)-f\left(L\right)|<ϵ\right]$.$\underset{x\to a}{lim}g\left(x\right)=L$ so $f\left(\underset{x\to a}{lim}g\left(x\right)\right)=f\left(L\right)$. spremani0r 2022-09-13

## Need to find $f\left(x{\right)}^{\prime }$ while $f\left(x\right)=ln\left(x+\sqrt{{a}^{2}+{x}^{2}}\right)$I have $f\left(x{\right)}^{\prime }=\frac{1}{\left(x+\sqrt{{a}^{2}+{x}^{2}}\right)}\cdot \left(1+\frac{2x}{2\sqrt{{a}^{2}+{x}^{2}}}\right)$, but can't simplify.I want get $\frac{1}{\sqrt{{a}^{2}+{x}^{2}}}$ Spactapsula2l 2022-09-12 nar6jetaime86 2022-09-12