What is Inverse function: Definition and 197 Discussions
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.
Hi there!
I'm back again with functions over matrices.
I have a function f : M_{n\times n} \to M_{n\times n} / f(X) = X^2.
Is valid the inverse function theorem for the Id matrix? It talks about the Jacobian at the Id, but I have no idea how get a Jacobian of that function. Can I see that...
Homework Statement
Problem: Given C is the graph of the equation
2radical3 * sinpi(x)/3 =y^5+5y-3
Homework Equations
(1) Prove that as a set
C= {(x,y) Exists at all Real Numbers Squared | 2radical3 * sinpi(x)/3 =y^5+5y-3
is the graph of a function differentiable on all real...
I can derivate x(y) wrt y using the derivative of y(x) wrt x, follows the formula: \frac{dx}{dy}=\frac{1}{\frac{dy}{dx}} until same the 2nd derivative (taking the 2nd diff form of x and deriving wrt to x):d^2x=\frac{d^2 x}{dy^2} dy^2 + \frac{dx}{dy} d^2y \frac{d^2x}{dx^2}=\frac{d^2 x}{dy^2}...
Homework Statement
Prove/Disprove following function being one-to-one.If yes,find its inverse.
g(x)=x-\frac{1}{x},x>0
Homework Equations
The Attempt at a Solution
My tutor said that it is one-to-one,but I found that the are two solutions for g-1(x).
Are there any mistakes...
Hi, All:
Let ## f: X → Y ## be a differentiable map , so that ## Df(x)≠0 ## for all ##x## in ##X##. Then the inverse function
theorem guarantees that every point has a neighborhood where ##f ## restricts to a homeomorphism.
Does anyone know the conditions under which conditions a map like...
can a function that's not inversable be inversible in certain interwalls. is it ok to say its inversable in this specific intervall or can't the function ever be called inversible?
Hi
I have a question regarding differentiation of inverse functions that I am not capable of solving. I want to prove that
\frac{\partial}{\partial y} h_y(h^{-1}_{y_0}(z_0))\bigg|_{y=y_0} = - \frac{\partial}{\partial y} h_{y_0}(h^{-1}_{y}(z_0))\bigg|_{y=y_0},
where
h_y(x) is...
Homework Statement
Find inverse of each.
1. y<x+1
2. y=2x/(x-2)
Homework Equations
Switch y and x?
The Attempt at a Solution
For 1. I switched y and x, so x<y+1. Do I have to switch the sign also?
For 2. I switched y and x, so x=2y/(y-2). But I have to express the inverse...
Homework Statement
Suppose f is a function defined on a set ##S## in ##ℝ^n## and suppose ##Q## is a limit point of ##S##.
If ##f(P) → 3## as ##P → Q## prove from first principles that ##\frac{1}{f(P)} → \frac{1}{3}## as ##P → Q##.
Homework Equations
The Attempt at a Solution...
Homework Statement
Let p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=(\lambda-x_1)(\lambda-x_2)(\lambda-x_3) be a cubic polynomial in 1 variable \lambda. Use the inverse function theorem to estimate the change in the roots 0<x_1<x_2<x_3 if a=(a_2,a_1,a_0)=(-6,11,-6) and a changes by \Delta...
Let $p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=(\lambda-x_1)(\lambda-x_2)(\lambda-x_3)$ be a cubic polynomial in 1 variable $\lambda$. Use the inverse function theorem to estimate the change in the roots $0<x_1<x_2<x_3$ if $a=(a_2,a_1,a_0)=(-6,11,-6)$ and $a$ changes by $\Delta...
Homework Statement
Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous.
Homework Equations
I have no idea.
The Attempt at a Solution
I seriously have no idea. This is for my analysis course, and I'm not...
Homework Statement
Find the inverse function of y= x/2 - 5/2x
Homework Equations
The Attempt at a Solution
I've tried to manipulate the equation to find x(y) without any sucsses.
Hi
Does anyone know if there is a relation between the Fourier transform of a function and the Fourier transform of the inverse function
in summary
FT[f(x)] ?= FT[f-1(x)]
Thanks!
Homework Statement
There were other questions before this one but i solved them all.
Find the inverse function of f(x)=arctan(\sqrt{1+x^{2}}-x) for every x in the interval ]0,pi/2[ .That's the interval that I found when counting f(R) because f is a bijection from R to f(R). Hence...
Homework Statement
The function f(x) has an inverse function, g(x). Find g'(5).
Homework Equations
f(x) = x^5 + 2x^2 + 2xThe Attempt at a Solution
I don't see how I can possibly find the inverse of this function. So I opted to use the derivative rule for inverses.
f'(x) = 5x^4 + 4x + 2 5...
Homework Statement
Let g(x) = (e^x - e^-x)/2. Find g^-1(x) and show (by manual computation) that g(g^-1(x)) = x.
Homework Equations
g(x) = (e^x - e^-x)/2
The Attempt at a Solution
I get the inverse = ln[ (2x + sqrt(4x^2 + 4) ) / 2 ]
How do I proceed?
Homework Statement
Homework Equations
The Attempt at a Solution
for 24. which is a quadratic function, i understand that you have to exchange the positions of x and y and solve from there...but my question is the inverse function of that quadratic for 24 is a negative number...
Consider the function f: R-->R where F(x)=
x, if x<1
x², if 1<x<9 (read x <or equal to 1 and x< or equal to 9)
27*sqrt(x), if x>9
I know that f is not differentiable at x=1 and x=9 because the derivative at each one of these points don't exist once the lateral limits don't coincide.(Is...
Hello
how can we know with the definition of inverse function
this function is inverse function or not?
?
another questions.
this function is inverse function because
but if we have this function
we can't do the same approach.why?(and it isn't a inverse function)...
Hi, so this isn't a question, it's just an example that they've given, but I don't understand the explanation given.
You have :
y = x^2 - 4
x = y^2 - 4
y^2 = x + 4
y = ± sqrt(x+4)
I don't get why there is a ± symbol there. My book says that it's necessary because there are two...
I don't understand why all authors of this proof assume that Df_a = id_n, how doesn't this destroy generality?
For example, see https://www.physicsforums.com/showthread.php?t=476508.
The λ in his post (and the post he quotes) is always Df_a (its not stated in that post, but in the book and the...
Homework Statement
Let f(x) = sinh(x) and let g be the inverse function of f. Using inverse function theorem, obtain g'(y) explicitly, a formula in y.
Okay the Inverse function Theorem says (f^-1)'(y) = 1/(f'(x))
If f is continuous on [a, b} and differentiable with f'(x)\neq0 for all x\in[a...
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...
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
Determine if the following function is invertible.
If it is, find the inverse function.
f (x) = -sin(-x)
-∏ < x < ∏
The Attempt at a Solution
f(x) = -sin(-x)
y = -sin(-x)
-(sin y)^-1 = -x
(sin y)^-1 = x
when x = -∏, y = 0
when x = ∏, y = 0
f^-1(x)...
I'm having a little trouble with something so I am wondering,
If f is a continuous 1-1 mapping from an open set (a,b) into ℝ then is its inverse function g continuous at all points of the image of f?
My argument is that g(y) is in (a,b) for all y in the image of f, and g(y)=x for some x in...
Homework Statement
Given the exponential function and its inverse, where a > 0
Exponential Function:
f(x)=a^x
Inverse function:
f^{-1}(x)=log_ax
a) For what values of a do the graphs of f(x)=a^x and f^(-1)(x) intersect?The Attempt at a Solution
I have no idea how to start this question.
Off...
Homework Statement
I'm trying to find the inverse function of the following.
f(x) = [x^2 - 9]^0.5
x <= -3
The Attempt at a Solution
In the book I'm using it gives the inverse function as;
y = -(x^2 + 9)^0.5
[0, +∞)
Is this correct, as I'm getting an inverse...
I am self-studying Calculus and tried to solve the following question:
Homework Statement
Suppose that the function f is continuous and increasing in the closed interval [a, b]. Then
(i) f has an inverse f-1, which is defined in [f(a), f(b)];
(ii) f-1 is increasing in [f(a), f(b)];
(iii) f-1...
My contention is that you cannot apply the Inverse Function Theorem to this problem because there is a point in the interval at which the derivative is zero.
At x = pi/2, f ' = 0.
http://i111.photobucket.com/albums/n149/camarolt4z28/Untitled.png
Homework Statement
Let, f(x) = x^3 - 3x^2 - 1 , x => 2 . Find the value of df^-1/dx at the point x=-1=f(3)Homework Equations
The definition states; "If f has an interval I as domain and f'(x) exists and is never zero on I, then f'(c) is diff at every point in its domain. The value of (f^-1)' at...
Homework Statement
Let f:A→B be a bijection and let f-1 :B→Aitsinverse. If Γ ⊂ A×B is the graph of f, show that the set, Γ′ ⊂ B×A defined as Γ′ = {(b, a)|(a, b) ∈ Γ} is the graph of f−1.
Homework Equations
The Attempt at a Solution
So far i have proved that the second projection of...
f(x) = (3 - e^(2x))^(1/2)
y^2 = 3 - e^(2x)
-(y^2 - 3) = e^(2x)
ln(-(y^2 - 3) ) / 2 = x
What am i doing wrong?
In the back of the book it says...
ln(-(x^2 - 3) ) / 2 = y
I'm trying to work out the method of getting the inverse function of
f(x) = x^2 + x I already know the inverse but I would like to know the method used to obtain it. So far I have:
Made f(x) = y:
y = x^2+2
And then made y = x and x = y:
x=y^2+y
And then I did this but I'm not sure if it's...
Homework Statement
f(x)=(x-3)^2 -1 for x ≥3
Homework Equations
The Attempt at a Solution
I am having difficulty grasping the concept of changing the greater than or equal to part of the equation above to it's inverse form. If for x it says x≥3 then how would that statement be...
is there a way to explicitly express the chi-squared inverse function?
when programming it, I have had to resort to a guessing system where I find a chi value that is too low and too high, and evaluate the chi-squared CDF to reset the high and low points iteratively until it is within a...
Hello,
I'am suffering with the theoretical background. My course state the follow thing:
D(f-1(y))=1/D(f(x)).
So: f-1(y) is the inverse function of f(x), this means that the argument of f-1(y) is y!
Example: y = f(x) = x^2 => f-1(y): x = y^2. Am I correct with is one ?
The chain...
Implicit => inverse function theorem (urgent due to exam, please help)
Homework Statement
Prove the inverse function theorem, knowing the implicit function theorem.
Homework Equations
The statements of both theorems... Can't think of much else.
The Attempt at a Solution...
I know it is impossible to apply inverse function theorem on a function from R^n to R^m because the Jacobian is not a square matrix.
Is there any other reason why this is impossible?
I am studying the multi variable Inverse Function Theorem and the Implicit Function Theorem. I think my brain is rebelling against understanding them and I would appreciate if someone here could explain the two theorems semi rigorously as well as explain when they are used, and why they are...
Homework Statement
Let f \in C1(Rn) be a function such that f(0) = 0 and \delta1f(0) is nonzero. (\delta1 means partial derivative with respect to x1)
Show that there exist neighbourhoods U and V of x=0\in Rn and a diffeomorphism g:U->V such that f(g(x)) = x1 for all x = (x1,...,xn) \in U...
Homework Statement
What is the inverse function of a bi-exponential function like the following:
Y=A*exp(B*X)+C*exp(D*X)
Homework Equations
Y=A*exp(B*X)
The Attempt at a Solution
If it is a single exponential function, i can take log on both sides to get inverse function. But when...
can a function in ONE dimension have NO inverse ?? i mean
if given the inverse function f^{-1} (x) = g(x) + \sum_{k=-N}^{k=N}c_{k}exp(ixlogk)
the first function g(x) is an smooth function , the last Fourier series is a 'noise correction' t o this function g , N is a big but finite...
let be the Fourier expansion of the function
f(x) = \sum_{m=-\infty}^{m=\infty}c_{m} exp(imx)
valid on the interval (-1,1) , from this can we obtain the inverse function
f^{-1} (x) by reflection of the Fourier series through the line y=x ??