0
votes
2answers
37 views

Getting inverse of polynoms with trigonometric functions

I'm trying to get the inverse of $$f(x) = \cos(x) + 3x$$ I tried it by definition of $\cos(x)$ with no luck: $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+...$$
1
vote
3answers
33 views

Simple inverse function of $\frac{1-2x}{1+x}$

Just started learning about inverse functions, and got stuck on this one: $$f(x) = \frac{1-2x}{1+x}$$ So I tried multiplying by $(1+x)$ on both sides and got $y+yx = 1-2x$ but that doesn't seem to ...
0
votes
1answer
52 views

To find the inverse of an implicit function

I have a function $t(f)$ here: $t(f) = T(sin(2\pi f/B)/2\pi + f/B) $ for $[-B/2 \le f \le B/2]$. $B$ and $T$ are constants. How to find the inverse of this function that is $f(t)$ using numerical ...
3
votes
2answers
54 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
0
votes
3answers
43 views

What would be the inverse function for the following condition?

What would be the inverse function condition for the above question.
2
votes
0answers
38 views

Determining $f^{-1}(3)$ without knowing $f^{-1}(x)$ but given $f(1)=3$ and $f'(x)>0$.

I have a continuous function $f(x)$ and I want to find $f^{-1}(3)$, but I can't find $f^{-1}$ directly. I know that $f(1)=3$ and $f'(x)>0$ for all x. Because the function is continuous and always ...
3
votes
8answers
202 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
0
votes
0answers
12 views

The inverse of a Moment generating function

The moment generating function of $X$ is $M_X(t) = \mathbb{E}[e^{tX}] = \int e^{tu}f_X(u)du$ where t is a complex variable and $f_X$ is the density of X. The cumulant generating funtion of $X$ is ...
0
votes
2answers
38 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 ...
2
votes
2answers
56 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
57 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
63 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
45 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
26 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
73 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
42 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
61 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
54 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
22 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
52 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
58 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
44 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
32 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
80 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
86 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
58 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
28 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
98 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
88 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
30 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
38 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
75 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
85 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
95 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
16 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
122 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
27 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
44 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
47 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
60 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
78 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
112 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
37 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 ...