0
votes
2answers
36 views

How to show the surjectivity of $f(x)=x^5$ on $\mathbb R$?

Sasy $f:\mathbb R\to\mathbb R$ define by $f(x)=x^5$ This is definitely injective as $x_1^5=x_2^5 \implies x_1=x_2$ I say it is surjective because for all really $x$ there is all real $y$, $x \in ...
-1
votes
0answers
21 views

Equation Inverse vs Solution Domain [closed]

Is finding the solution in a domain of a equation the same as doing the inverse operation? Example: Domain of $x$: $[1,2,3]$ $5x - 1 = 9$ ? I would do the inverse operation, but my teacher tells ...
2
votes
2answers
54 views

If a one-to-one function's inverse is the same what must be true of the graph of f?

As a followup to this question. I'm trying to determine what must be true of the graph of $f$ in these cases. I've examined the two functions $f(x)= x$ and $f(x)= \frac{1}{x}$ and I'm not seeing any ...
1
vote
2answers
54 views

Why does the square root of an inverse function turn negative?

For example, $$f(x)=x^2$$ $$y=x^2$$ $$-\sqrt{x} = f^{-1}$$ Why does $\sqrt{x}$ become negative? Edit: Sorry for all the confusion, I will state the problem on my textbook and the solution. ...
1
vote
4answers
62 views

If a function $f$ is decreasing on its domain then would its inverse be increasing or decreasing?

I have a question concerned the inverse of a function $f$ and the sign of its derivative. If we are given a function $f$ that is decreasing on its domain, would its inverse $f^{-1}$ be increasing or ...
0
votes
1answer
43 views

Help to prove the existance of a function

Let $f:X \rightarrow Y$ be a function. Prove that there exists a function $g:Y \rightarrow X$ such that $f \circ g = I_Y$ if and only if $f$ is a surjection. I need help on proving the following: ...
0
votes
0answers
23 views

Can anyone give the equation of the inverse of radial projection from a tetrahedron to sphere?

$(x,y,z) \mapsto \bigg(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}} \bigg)$ This is the equation of the radial projection. I need the inverse of this ...
1
vote
1answer
72 views

It $f(x)=x+\sin x$, then can we find $f^{-1} (x)$?

We have a bijective function $f(x)=x+\sin x$. So what is $f^{-1} (x)$? Let $f^{-1}(x)$ be $g(x)$. Suppose we have to find $g\left(\dfrac{\pi}{6}+\dfrac{1}{2}\right)$ and ...
2
votes
3answers
38 views

Is the inverse of this function unique

Let $f$ be a function from any set(Say $K$) to any set (say $P$) Now: $f(x)=2x+1$ My question:Is it necessary that the inverse of the function is $\frac{x-1}{2}$? This is a problem given in my ...
1
vote
1answer
60 views

finding exact value of $\sec^{-1} 5$

Find the exact value of $\sec^{-1} 5$ (decimal answer). I know that $\sec^{-1}5=\cos^{-1}\dfrac{1}{5}$, but I don't know how to proceed from here. I drew a right triangle with sides $1$ and $5$ ...
2
votes
3answers
49 views

Is fractional inverse of a function a known thing?

I know there's fractional Fourier transform, fractional derivative, maybe some other transformations generalized from being discrete to continuous. Now I wonder if there's any way to generalize a ...
0
votes
1answer
21 views

Inverting complicated function (possibly using secant root finder)

So I have the following equation from the 2002 paper "A Rapid Hierarchical Rendering Technique for Translucent Materials" http://graphics.ucsd.edu/~henrik/papers/fast_bssrdf/fast_bssrdf.pdf Here is ...
1
vote
3answers
44 views

Why rotating a function around line $y=x$ gives an inverse of this function?

So I'm trying to read through a book on calculus on my own and there is a statement that if we have a graph of some function $y=f(x)$ and this is an injective function, then rotating it around the ...
0
votes
1answer
81 views

What is the inverse function of $x-\log(\log(x))$?

What is the inverse function of $f(x)=x-\log(\log(x))$? If we restrict the domain to e.g. $x\in[2,+\infty[$, the function should have an inverse, but I am unable to compute it.
1
vote
1answer
53 views

When inverse functions are helpful?

I pass some colloquiums to find inverse functions. But still can't understand the real help of them. Only one real world example come to my mind: converting units of measurement (but those convertions ...
0
votes
2answers
42 views

Inverse functions problem

There are two functions $f\colon\mathbb Q \to \mathbb Q \setminus \{-1\}$ and $g\colon\mathbb Q \to \mathbb Q \setminus \{1\}$. $$g(x) = \frac{f(x)}{f(x)+1}.$$ Prove that if there is a inverse ...
0
votes
1answer
27 views

Inverse image of an element in co-domain but not in range?

Sorry, quite new to this. I have a question that contains the image below of $g:X\rightarrow Y$ and it is asking for the inverse image of $u$. Am I correct in thinking that the answer is $\emptyset$? ...
2
votes
4answers
78 views

Finding the inverse of $f(x)=|x|-2$

How would I find the inverse of the function $f(x)=|x|-2$? I have swapped $x$ and $y$, and tried to isolate $y$, reaching up to $x+2=|y|$ Whenever I see absolute values, I always break the problem up ...
2
votes
1answer
83 views

Minimum Argument Difference to Make the Lower Bound > the Upper Bound

Assume $g$ is a function that grows asymptotically as $$ g(n) \in\frac n {log(n)} + O(\sqrt n),\,n \in \Bbb N\tag1 $$ I wish to find $h(n)$ such that $$ g(n) \le g(n+h(n)). $$ i.e. Given the bounds ...
2
votes
2answers
57 views

Find the inverse of $f(x) = (x+1)/(x-8)$

Find the inverse of this function: I have gotten this far: $x = y+1/y-8$ $x(y-8) = y+1$ $x(y-8)-1=y$ $xy-8x - 1 = y$ I think I went backwards?
1
vote
2answers
27 views

Inverse function (basic algbra math)

Consider the following function: $f(x) = {1 / (x-6) }$ Find a formula for the inverse of the function. Here is what have so far? $y = 1/(x-6)$ ---> $ x = 1/(y-6) $ But my embarrassing problem is ...
2
votes
5answers
96 views

How to find the inverse of $f(x) = \frac{x+2}x$?

What approach would be ideal in finding the inverse of $f(x) = \frac{x+2}x$?
2
votes
2answers
45 views

Is $\sec^{-1}(\sec(\pi/2)) = \pi/2$?

I think it shouldn't be defined as $\pi/2$ is not in the range of the function $\sec^{-1}(x)$ Wolfram confused me by giving the answer as $\pi/2$ : Link But it mentions on another page that $\pi/2$ ...
0
votes
1answer
77 views

What are some practical uses of functions? [closed]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?
1
vote
2answers
29 views

Problem inverting a function

I have this function: $$v(t)=\sqrt{\frac F c} \tanh \left(\frac{\sqrt{Fc}}{m} t \right)$$ I can visually see that t=6.3 when v=27.8, so why don't I get t=6.3 upon putting v=27.8 in this supposedly ...
0
votes
2answers
24 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
1
vote
2answers
34 views

Inverse modulo function

How can we calculate the inverse of a modulo function, now I have a problem given me $f(n)=(18n+18)\mod29$, need find inverse of $f(n)$ ? how is the process to do it?
3
votes
0answers
74 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
1
vote
2answers
84 views

Tangent line of the inverse function of $y = e^x + x$

I've been sitting on this problem for a while, hopefully you guys could give me a lead on what the hell is going on :) Let $f(x) = e^x + x$ Find the tangent line to $f^{-1}(y)$ (the inverse ...
0
votes
1answer
19 views

Linear Growth Model

I have a problem where I have been given that $r(t)=at+b, 0 \leq t \leq \frac{100-b}{a}$. I have then been asked to find $t(r)$. Is this simply finding the inverse of $r(t)$?
1
vote
2answers
30 views

How can I find the inverse of this function? [closed]

Can anyone help me find the inverse of this function? $$y=\frac{x}{2}-\frac{x^2}{16}$$
1
vote
1answer
89 views

Inverse function theorem question - multivariable calculus

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$ 1) ...
1
vote
1answer
15 views

Arc Tangents and Equation

For one of the problems in my book, it requires you to put the arc tangent into the 2piK equation and solve for the arc tangents and lie in [0,2pi]. For: arctan(117)+piK the answers are 1.5622 and ...
1
vote
1answer
95 views

Inverse Trig Functions with Double Angle Formulas

I am studying for a quiz tomorrow and one of the sections I am studying involves rewriting quantities as algebraic expressions of $x$. One of the problems I am having trouble with is: $$\sin ...
-1
votes
4answers
26 views

Prove Inverse Function [closed]

Consider the function $f:\Bbb R\times\Bbb R→\Bbb R\times\Bbb R$ defined by $$f(x,y)=(x+y,x-y)$$ This function is invertible. Show that the inverse function is given by $$f^{-1} (a,b)=\left( ...
1
vote
0answers
19 views

Can we describe an original and inverse equation with one function?

Let us say we have two real values, 1 and x. I want to determine the absolute value of the difference or their sum between 1 and x without specifying whether I am dealing with 1 - x or 1 + x For ...
1
vote
2answers
42 views

Show that $ (f^{-1})^{-1}=f $

$$ f:X\to Y $$ $f$ is invertible, show that $(f^{-1})^{-1}=f$ Here it is not given that how the function is defined, so I think that making equations and solving them will not help me. So I have ...
1
vote
2answers
46 views

Find the inverse of a function.

$$ g:[-1,1] \to \mathbb R\\ g(x)={\frac{x}{x+2}} $$ $f:[-1,1] \to$ range of f. Find the inverse of $ f.$ $\forall y\in \text{range of }g$ there exist some ${\frac{2y}{1-y}}\in [-1,1]$ such that ...
0
votes
4answers
71 views

$f:\mathbb R \to (0,\infty)$ defined by $f(x)=e^x$. Describe its inverse.

How do I go about describing it? Well first is the inverse $e^{-x}$ or $\ln(x)$? Additionally, since I have no clue how to solve these problems as I am probably overthinking them... $f:\mathbb R\to ...
0
votes
0answers
50 views

How to calculate the inverse of a known optical distortion function?

Assume I have the following lens distortion function: $$ x' = x(1 + k_1r^2 + k_2r^4) \\ y' = x(1 + k_1r^2 + k_2r^4) $$ where $r^2=x^2 + y^2$. Given the coefficients $k_1$ and $k_2$ I would need to ...
1
vote
1answer
75 views

Express parametric curve as graph of a function

I have a parametric curve in $\mathbb{R}^2$ given by $$ t\mapsto f(t)\left(\begin{array}{c}1\\1\end{array}\right)+\sqrt{-f'(t)}\left(\begin{array}{c}1\\-1\end{array}\right),\quad ...
3
votes
4answers
101 views

What is the proper way to find the inverse of a function?

I am a little confused on the subject of inverse functions and the methods used to do the transformation from function to inverse. How do you make an inverse? Just so i can avoid any ambiguity in my ...
1
vote
1answer
36 views

Find if the system $(x(t-1))^2 + x(t) +(x(t+1))^2 = y(t)$ is invertible

If there wasn't the $x(t)$ term, I could use $x(t) = x$ and $x(t) = -x$ to disprove invertibility, but I can't think of two functions that give the same $y(t)$ in this case. When I tried proving ...
0
votes
1answer
39 views

Finding the inverse of a function involving |x|

I need to find the inverse of the following function $ f:(-1,1) \rightarrow \mathbb{R} $ $ f(x) = \dfrac{x}{1-|{x}|} $ How do I deal with the absolute value here? Thanks
1
vote
2answers
75 views

What is the inverse of this function?

please help me to find out the inverse this function, $$f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}}$$ I know that, let $$y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$ and if I find $x=\cdots$ then that is the ...
2
votes
1answer
35 views

Why is there a left inverse for an injective Function with the empty set as domain?

The fact that a function is injective is equivalent to the fact that there is a left inverse. Now consider $f:\mathbb{∅}\to \mathbb{A}$ where $\mathbb{A}$ is non-empty. Wouldn't the left inverse be ...
1
vote
2answers
1k views

Inverse of function, containing a fraction

This is basic, I know, but I cannot seem to come up with the right answer. Find the inverse of the function: $$f(x)= \frac3{x+1}$$ My steps: 1. Convert f(x) to y $$y = \frac3{x+1}$$ Switch places ...
0
votes
4answers
84 views

Let $f: A\rightarrow B$ and $g: B\rightarrow C$ be invertible maps, show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

I am working on the following problem for my abstract algebra class, and I wanted to get some feed back to see if I am on the right track. Here is what I have so far. Let $f: A\rightarrow B$ and $g: ...
2
votes
1answer
48 views

Inverse of $f(x)=\sin(x)+x$

What is the inverse of $$f(x)=\sin(x)+x.$$ I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet. What about $$f(x)=\sin(a \cdot x)+x$$ where ...
0
votes
0answers
62 views

Uniform continuity of inverse in only one variable

Let $f:[0,1]\times[0,1]\to \mathbb{R}$ be a (uniformly) continuous functions. Denote the image of $f$ by $D_f:=\{(x,y): x\in[0,1] , 0\leq y \leq f(x,1)\}$ $f$ is such that the section $f_x$, i.e. the ...