In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, i.e., g(y) = x if and only if f(x) = y. The inverse function of f is also denoted as
f
−
1
{\displaystyle f^{-1}}
.As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we would undo each step in reverse order. In this case, it means to add 7 to y, and then divide the result by 5. In functional notation, this inverse function would be given by,
g
(
y
)
=
y
+
7
5
.
{\displaystyle g(y)={\frac {y+7}{5}}.}
With y = 5x − 7 we have that f(x) = y and g(y) = x.
Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse, it must have the property that for every y in Y, there is exactly one x in X such that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f.
Homework Statement
Suppose f: A → B is a function. Show that f is surjective if and only if there exists g: B→A such that fog=iB, where i is the identity function.The Attempt at a Solution
Well, I believe for a rigorous proof we need to use the axiom of choice, but because I have never worked...
I was reading <crackpot link removed> and was wondering if the inverse cube law for magnetic force still applied for situations where the object being attracted isn't another magnet itself? E.g. if there is an electromagnet attracting an iron nut is the rule still inverse cube and not inverse...
Homework Statement
y=x+(1/x) at y=17/4
Homework Equations
The Attempt at a Solution
y^-1: x=y+(1/y)
differentiate: 1=y'+ln(y)y'
1=y'(1+ln(y))
y'=1/(1+ln(y))
put that over 1: 1+ln(y)
plug in y: 1+ln(17/4)
=approximately 2.447
Consider a function f(x) and its inverse g(x).
Then (f \circ g)(x) = x and (g \circ f)(x) = x
Are both these statements separate requirements in order for the inverse to be defined? Is it possible that one of the above statements is true but not the other? If so, could I see an...
Homework Statement
This is from a physics problem, but my question is more mathematically oriented.
After working through the problem, I arrive at the last step.
Sin(x)=.967
The question says that there are two possible angles for x.
The Attempt at a Solution
arcsin(.967)...
My analysis text defines inverse functions only for bijections.
But y = e^{x} is not bijective, so according to my book it's inverse ( ln x ) wouldn't be defined? Am I missing something or is my textbook just plain wrong?
I use the text by Bartle and Sherbert.
Thanks!
BiP
Homework Statement
The Attempt at a Solution
Suppose we take an arbitrary polynomial in P_2 (R), call this a_0 + a_1 x + a_2 x^2
T(a_0 + a_1 x + a_2 x^2) = (a_0, a_0 + a_1 + a_2, a_0 - a_1 + a_2)
Now, I was under the impression that I could construct a matrix for T by showing what...
Homework Statement
Find the derivative of sec^{-1}(\frac{\sqrt{1+x^{2}}}{x})
Homework Equations
sec^{-1}=\frac{U'}{U\sqrt{U^{2}-1}}
The Attempt at a Solution
U'=-\frac{1}{x^{2}\sqrt{1+x^{2}}}
U\sqrt{U^{2}-1}= \frac{\sqrt{1+x^{2}}}{x^{2}}
Therefore the derivative is...
Homework Statement
Find derivative of tan^{-1}(\frac{3sinx}{4+5cosx})
Homework Equations
deriviative of tan^{-1}=\frac{U'}{1+U^{2}}
The Attempt at a Solution
I found U'= \frac{12cosx+15}{(4+5cosx)^{2}}
1+U^{2}=1+\frac{9sin^{2}x}{(4+5cosx)^{2}}
I think my components are correct but my...
Homework Statement
Find the derivative of:
sqrt(x^2-4)-2tan^-1{.5*sqrt(x^2-4)}
Homework Equations
U'/1+U^2
U'=x/2sqrt(x^2-4)
1+U^2=x^2
The Attempt at a Solution
I combined the above components but my answer is incorrect. I feel that I might have the wrong answer for...
Homework Statement
Hi there I'm trying to solve this question:
Homework Equations
The Attempt at a Solution
I figured i should just multiply them together and show that you get the identity matrix, but I'm having trouble cancelling out some of the terms. I'm not sure if I...
Using the laplace transform, find the solution to the differential equation:
y'' + y' + y = 0 , y(0)=0, y'(0)=1
Using the laplace transform and its properties I end up with:
f(s) = 1/(s2+s+1)
How can I find the inverse of this/ does anyone know the inverse of it?
Setting y=eax I got a...
Homework Statement
Hi. I found in the answears that the inverse of function f(x)=3-\sqrt{x-2} is f^{-1}(x)=(3-x)^{2}+2 only if we restrict it to {x:x\leq3}. I understand that the restriction is needed because the found inverse is a parabola (and thus not one-to-one function).
My general...
Derivative of inverse function at x=0 [SOLVED]
Homework Statement
Let f(x) = x + Ln(x+1), x > -1
Find \frac{d}{dx} f^{-1} |_{x=0}; Note that f(0) = 0
Homework Equations
(f^{-1})'(x) = \frac{1}{f^{'}(f^{-1}(x))}
(or)
\frac{dx}{dy} = 1/\frac{dy}{dx}
The Attempt at a Solution...
Suppose that $\displaystyle f(x)=\frac{ax+b}{cx+d}$. What conditions on $\displaystyle a,\ b,\ c,\ d$ are necessary and sufficient in order that $\displaystyle f(x)$ coincide with its inverse function.
My attempt:
$\displaystyle...
Homework Statement
The Attempt at a Solution
So I observed:
T(B) = λB
T-1(B) = λ'B
Also,
T-1(T(B)) = λ'λB = B
This implies,
λ'λ = 1
And so, there should be a relation
λ = \frac{1}{λ'}.
Is that right?
Homework Statement
How do I find the inverse of these functions step by step?
y= e^-x^3
y= sin(1/x)
I know the solutions but I don't know how to work with these two functions. Does anyone know the steps to finding the inverse of these?
Hi all
I have been working on some unique solutions to advection-diffusion type problems.
One inverse Laplace transform that I seem to continue to encounter is the following:
Inverse Laplace[F(s)] where F(s)=[(1/(((s-α)^2)+β)*exp(-x*sqrt(s/D))]
In their classic 1959 text, Carslaw...
Why are derivatives of inverse functions important?
My students are giving me questions like:
When would using the theorem be useful? Can't you just find the inverse function and take its derivative?
I'm sure many of you know the type of question: "Who cares?"
My answers are that the theorem...
Homework Statement
What is the inverse of matrix A?
A=
(1) (2) (1)
(2) (-1) (2)
(1) (2) (1)
Homework Equations
The Attempt at a Solution
The determinant works out to be 0
The inverse is 1/determinant x adj(A)
Therefore the inverse is 1/0 x adj(A)
So is the...
Let T: V-->W be a surjective linear transformation and let X be a subspace of W. Assume tat Ker(T) and X are finite dimensonal.
Prove that T^-1 (X) = {v | T(v) is in X} is a subspace of V.
Ok I absolutely suck at showing things are a subspace of something...I don't even know where to...
Hello everyone,
I've been trying to figure out how to obtain the multiplicative inverse of an integer in Zn in Mathematica but I haven't found a way. Is there a way to do this anyone can help me with?
Homework Statement
This is an integration of an inverse trig function.
I don't see how they go from 1/2 to 1/4. I understand how they get the 1/2, du = 2dx, divide both sides by 2, but where does the 1/4 come from?
How does one use IFT to find the inverse of a function? I thought it was something like \int \frac{dx}{df(x)}dx. But that doesn't work with f(x)=x^2:\int \frac{dx}{2x}=\frac{1}{2} \log{x} \neq f^{-1}(x).
Assume that there is a original wall which does not have any defect inside. After an excitation of heat at one side of the wall, Ti is 37°C and To is 30° C. Thermal capacitance is assume 1. So the heat transfer is 7.
For another condition, i want to find out if the wall has air pocket inside...
Homework Statement
tan-1[2x/(1-x2)]+cot-1[(1-x2)/2x]=2π/3
2. The attempt at a solution
tan-1[2x/(1-x2)]+cot-1[(1-x2)/2x]=2π/3
tan-1[2x/(1-x2)]=2π/6
take tan on both sides
2x/(1-x2) =sqrt(3)
quadratic equation so it should have 2 solutions(sqrt(3) and sqrt(1/3)).But this question...
Hey everyone. Sorry to post another topic, but i thought this would be easy to find on google, to help me with , but i can't find it.
All I am really lookin for is someone to let me know if my answers are right, and if not, then how i can fix them :)
Anyways, i was asked to graph...
IVP Laplace Transform Problem -- Tricky Inverse Laplace Transform
Homework Statement
Solve x"+x'+x=1, given x(0)=x'(0)=0
Homework Equations
The Attempt at a Solution
Plugged in transforms: s2*Y(s)-s*y(0)-y'(0)+s*Y(s)-y(0)+Y(s)=1/s
Plugged in initial value points, simplified...
Homework Statement
Hello all,
Having difficulty with this one question that involves complex roots. Here it is:
F(s)=\frac{s+3}{s^3+3s^2+6s+4}
I tried two different ways to tackle it. First method I divided it right away:
F(s)=\frac{s+3}{s^3+3s^2+6s+4}\rightarrow{s^2+6-\frac{14}{s+3}}
Is there...
Why are Newtons law of universal gravitation,
F=G\frac{m_{1}m_{2}}{r^{2}}
and Coulombs law,
F = K_{e}\frac{q_{1}q_{2}}{r^{2}}
inverse square laws? I understand why they are inverse because the force decreases with distance but why is the distance, r, squared?
Thanks
AL
I feel kind of lame, but here's my situation:
We start with the operator g_{\mu \nu} \Box - \partial_{\mu}\partial_{\nu} and convert to momentum space to get -g_{\mu \nu} k^{2} - k_{\mu}k_{\nu}.
Apparently it's easy to see that this has no inverse?
I'm told that if it *did* it would be...
I am reading the proof of the Inverse Function Theorem in baby Rudin and I have a question about it. How does associating a function phi(x) (equation 48) with each point y tell us anything about if f(x) is one-to-one? I'll show the proof below. Also, if f'(a) = A, and f(x)=x2, what would A-1 be?
Homework Statement
find the derivative of the function
f(x)=arcsec(4x)
Homework Equations
I think this is a Relevant equations.
d/dx[arcsecu]=u'/(|u|(√u2-1)
The Attempt at a Solution
f'(x)=4/(|4|(√42-1)
=1/√15
I keep getting wrong in my online homework why? :confused:
Homework Statement
arctan(8x-8)=-1
Homework Equations
I'm sure what this part wants.
The Attempt at a Solution
tan[arctan(8x-8)]=tan(-1)
8x-8=tan(-1)
x=(1/8)(tan(-1)+8)
x=(1/8)(-(tan(1)+8)
x=
I am stuck on the last part since the homework says Simplify the above equation...
Homework Statement
Find the derivative of the compositional inverse of f(x) = sin(1/x) restricted to (1,∞). You may use without proof that sin(x) is differentiable with derivative cos(x).
Homework Equations
(f^{-1})'(y_0) = \frac{1}{f'(f^{-1}(y_0))}
The Attempt at a Solution...
Homework Statement
Find (f^{-1})'(a) of: f(x)=\sqrt{x^{3}+x^{2}+x+22} ; a=5.
Homework Equations
(f^{-1})'(a)=\frac{1}{f'((f^{-1})(a))}
The Attempt at a Solution
Well, I know to find an inverse: I need to set the equation equal to y, solve for x, then swap x and y. But I don't...
Homework Statement
The function f:x→ 4-x2 for the domain x≤0. Find the inverse of is denoted by f-1 and state the domain and range of f-1.
Homework Equations
Set equation to 0 and solve for x to find inverse. The D and R is going to be switched for the inverse..?
The Attempt at a...
I am having problem with the inverse transformation of a Fourier transformed function which should give the function itself.
Let
f=f(x) and let f be Fourier transformable (whatever that implies)
Let
\tilde{f}(k)=∫^{\infty}_{-\infty}dx e^{-ikx}f(x) (1)
then we should have...
Homework Statement
For α > 0, determine u(x) by the inverse Fourier transform
u(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ \frac{e^{ikx}}{ik+\alpha}\ dk
Homework Equations
The Attempt at a Solution
This seemed like a relatively simple residue problem. You just note that...
Homework Statement
∫ (x+2)dx/√(4x-x2)
Homework Equations
why was the -2 in -2(x-2) was ignored?
The Attempt at a Solution
so first i let u= 4x-x2
then, du=4-2x
= -2(x-2)
so to get (x+2) i equate it to (x-2)+4
so ...
∫ (x+2)dx/√(4x-x2) = ∫(x-2)+4dx/√(4x-x2)
= ∫...
This is insane, I am trying to revise the inverse of matrices and this one element is being really stubborn, please help.
Here is the matrix
3 -1 7
2 0 1
5 -2 6
I have transposed it
3 2 5
-1 0 -2
7 1 6
Now as for replacing the element of...
Homework Statement
The function
y=x^{2}+4x-6
has two inverses. What are they and which domains lead to these inverses?
Homework Equations
The Attempt at a Solution
y=x^{2}+4x-6
x=y^{2}+4y-6
y(y+4)=x+6
Not really sure where to go from here.
Homework Statement
(part of a problem)
Find the inverse Fourier of F(w) = (3jw+9)/((jw)^2+6jw+8)
where w is the angular frequency, w=2pi * f = 2*pi/T
Homework Equations
The fourier transfrom and its properties i guess.
Also the exponential FT common pair exp(-at)u(t) <-> 1/(jw+a)
where...
Homework Statement
L^{-1}\{\frac{1}{(s^2+4)^2}\}
Homework Equations
The Attempt at a Solution
I have no idea how to solve this. Any idea to being solving the problem would be appreciated.
Homework Statement
My problem is as follows:
find the inverse of
3x+1+\sin(x) with the domain [-\frac{\pi}{2},\frac{\pi}{2}]
Homework Equations
The Attempt at a Solution
for this would I just try to solve as normal by setting y=f(x) then using the fact that \arcsin(x) = y or is...
Suppose that a spaceship is fired into orbit from Cae Canerveral. Ten minutes after it leaves Cape, it reaches its farthest distance north of the equator, 4000 kilometers. Half a cycle later it reaches its farthest distance south of the equator (on the other side of the Earth, of course!), also...
Homework Statement
Greetings everyone! I am having problems getting started with this question, which asks:
Find the inverse of y = \frac{3-4e^x}{6-10e^x}
Homework Equations
The Attempt at a Solution
I'm usually pretty good with these kinds of problems, but for some reason I seem to be...