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.
Attempt : Let me copy and paste the problem as it appeared in the text. Please note that the given problem appears in part (b), which I have underlined in red ink in this way ##\color{red}{\rule{50pt}{1pt}}##
Clearly the domain is ##\boxed{\mathscr{D}\{f(x)\}...
Hi,
I am having a hard time trying to solve this question. How do I find the local inverse at x0?
f (x) = x^4 − 4x^2
Find an expression for f^−1 for f at the point x = −2.
Thanks a lot! I would really appreciate any help!!
Attempt : The domain of the function ##\sin(3x^2+1)## is clearly ##x\in (-\infty, +\infty)##. The values of ##x## go into all quadrants where the ##\sin## curve is positive and negative. Hence the range of the function ...
The problem goes as follows: Let ##M, N## be sets and ##f : M \rightarrow N##. Further let ##L \subseteq M## and ##P \subseteq N##. Then show that ##L \subseteq f^{-1}(f(L))## and ##f^{-1}(f(P)) \subseteq P##.
Obviously, I would simply use the definition of a functions inverse to obtain...
Hi everyone
This is the solution for the problem.
I don't understand how they got from
To
This was my attempt at a solution
I can't seem to get rid of one of the y terms and am left with one on each side.
Could someone explain the solution to me please?
Thanks
Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$
ok not real sure what the answer is but I did this (could be easier I am sure}
rewrite as
$y=(2x+1)^3$
exchange x and rename y to g
$x=(2g+1)^3$
Cube root each side...
Summary: Please see the attached problem and solution
The answer is 1/5. I have tried various solutions and cannot get 1/5.
What is my error?
[Moderator's note: Moved from a technical forum and thus no template.]
Summary: I am studying inverse functions and want to see a plot of an inverse function.
I hope this is an OK post here.
Lets say I have a function y = x^3 + x. This function has n inverse sine the derivitave is always positive and is a one on one function.
I can easily graph this function...
Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$$(A)\, \dfrac{2}{27} \quad
(B)\, \dfrac{1}{54} \quad
(C)\, \dfrac{1}{27} \quad
(D)\, \dfrac{1}{6} \quad
(E)\, 6$ok not sure what the best steps on this would be but assume we first find...
If I'm given a function ##f(x)##, say it has continuos first derivative, then I expand it as ##f(x + \Delta x) = f(x) + (df / dx) \Delta x##. If instead, I'm given ##f^{-1}(x)## how do I go about expanding it? Will this be just ##f^{-1}(x + \Delta x) = f^{-1}(x) + (df^{-1} / dx) \Delta x##?
Homework Statement
Find the inverse function of ##f(x) =x^4+2x^2, x>0##
Homework Equations
##f(f^{-1}(x)) = x##
The Attempt at a Solution
My only progress so far is
##x^4+2x^2 = x^2(x^2+2)##
Then I am stuck.
Since my progress is close to nothing so I don’t expect a complete...
Homework Statement
In calculus, I learn that the derivative of the inverse function is
g'(x) = 1/ f'(g(x))
Homework Equations
So..
The Attempt at a Solution
Can someone give me an example of where I need to know this, or is this just a math exercise. Is there a relatively simple physics...
Hello, I am trying to find the inverse of f(x) = x \div (x+4)
IIRC i need to replace f(x) with y and solve for x.
I've tried to do y = x \div (x+4) becomes y(x+4) = x then xy + 4y = x but I can't reduce the amount of x to one.
What am I doing wrong in this problem?
I'm currently carrying out an analysis on waveforms produced by a particular particle detector. The Nyquist-Shannon sampling theorem has been very useful for making an interpolation over the original sample points obtained from the oscilloscope. The theorem (for a finite set of samples) is given...
Let ##f:U\subset R^n\rightarrow V\subset R^n## be a biyective function of class ##C^m(m\geq 1)##, ##U## and ##V## are open sets in ##R^n##. I know from the inverse funtion theorem that when ##J(f)\neq 0## in a point of the the domain a local inverse exists, however, given the above conditions...
Homework Statement
##f:= tanh = \frac{e^x-e^{-x}}{e^x+e^{-x}}##
Prove that
##f^{-1}(x)= \sum\limits_{k=0}^{\infty} \frac{x^{2k+1}}{2k+1}## for all x in (-1,1)
The Attempt at a Solution
I also found the inverse function to be:
##f^{-1}(x)= \frac{1}{2}ln(\frac{1+x}{1-x})##
I tried working...
Hello,
I have tried the integral below with Mathematica and it gives me the following solution:
##\frac{d}{dc}\int_{z^{-1}(c)}^{1} z(x)dx = -\frac{c}{z'(z^{-1}(c))}##
I am not quite sure where it gets it from...I think it can be separated and with differentiation the first part will be zero...
Homework Statement
Suppose ##f(x) = x^5 + 2x + 1## and ##f^{-1}## is the inverse of function f. Evaluate ##f^{-1}(4)##
solution: 1/7
Homework Equations
##(f^{-1}(x))=\frac{1}{f'(f^{-1}(x))}##
The Attempt at a Solution
I attempted to use my calculator's solve function to get the solution of...
Homework Statement
Snell's law is:
$$\frac{\sin\theta_1}{c_1}=\frac{\sin\theta_2}{c_2}$$
$$\frac{c_1}{c_2}=n_{12}$$
Express ##\theta_2## as a function of ##\theta_1##
Find the largest value of ##\theta_1## for which the expression for ##\theta_2## that you just found is...
Homework Statement
Simplify:
$$\sin^{-1}(2\sin^{-1}0.8)$$
Homework Equations
Inverse sine: ##y=\sin^{-1}(x)~\rightarrow~\sin(y)=x##
$$\sin^2(x)+\cos^2(x)=1$$
The Attempt at a Solution
The inner parenthesis: ##\sin y=0.8## . In the drawing it's alpha's sine.
Now i double the α and the...
Homework Statement a.)y=sqrt(2^x -1) . I tried:
b.)y=log(sqrt(2^x -2)) and
c.)y=log^3 (2-sqrt(x)).
Homework EquationsThe Attempt at a Solution
x=sqrt(2^y -1)
x^2 = 2^y -1
2^y = x^2 +1
y=log2(x^2 +1)
y=2 log2(x+1) is that correct result?
regarding b and c I am just lost :/. Will appreciate...
Hello all
I was doing some approximation to solve another problem, but got stuck when trying to figure out a suitable inverse functions for this:
a = \frac{\cos x}{3x^2 - \pi^2}, where 0 \le x \le \pi.
What I need is the two functions x(a) at least near a \approx -0.086 \pm 0.01 but I'm not...
Homework Statement
Proof that: f has an inverse ##\iff## f is a bijection
Homework Equations /definitions[/B]
A) ##f: X \rightarrow Y##
If there is a function ##g: Y \rightarrow X## for which ##f \circ g = f(g(x)) = i_Y## and ##g \circ f = g(f(x)) = i_X##, then ##g## is the inverse...
Homework Statement
Let f(x) = 1−3x−2x^2 , x ∈ [−2, −1]. Use the Horizontal Line Test to show that f is 1–1 (on its given domain), and find the range R of f. Then find an expression for the inverse function f −1 : R → [−2, −1].
The Attempt at a Solution
I have already done the horizontal line...
Homework Statement
Solution set of the inequality (cot-1(x))2 -(5 cot-1(x)) +6 >0 is?
Homework EquationsThe Attempt at a Solution
Subs cot-1(x)=y
We get a quadratic inequality in y.
y2-5y+6>0
(y-2)(y-3)>0
Using the wavy curve method, the solution set is...
T(t) = Ts+(98.6 – Ts)e-kt
rewrite in the form t=g-1(T)
In trying to understand how to find the inverse of this but am having a hard time, please advise.
Thanks,
Kupkake303
Hello,
I need to find the inverse function of the following equation
y = a * ((exp(-b * x)) + (c * (1 - (exp(-b * x)))))
Where a, b and c are constants.
I have experimental points that fit to this equation and I want to use these values in the inverse funtion to linearize it.
I have tried...
Define $f: \mathbb{Z} \to \mathbb{Z}$ via $f(n) = n^2$ for all $n \in \mathbb{Z}$. Why does $f^{-1}(\left\{0,1,2\right\}) = \left\{0,-1,1\right\}$? The definition I'm using is $f^{-1}{(T)} = \left\{a \in A: f(a) \in T \right\}$ so we have $f^{-1}({ \left\{0,1,2\right\} }) = \left\{n \in...
if
then to prove an inverse of this exists the following has been done to show that it is one to one
what is the basis of equating the 2 square roots ?
Homework Statement
Hello,I have some problems with my Pre-Calculus homework. The task is:
You get paid 8$ per hour plus 0.85$ per unit you produced.
1.Set up an equation for it.
2.Find the inverse function.
3.What does each variable in the inverse function mean?
Homework Equations
See below...
I noticed the scan was cut off on the second image at the bottom right, but I came up with x= 31/5
My first test in Calc I begins tomorrow and I want to know that I'm headed in the right direction. I think I understand to some extent how logarithms can be expanded and condensed though I'm...
I have a set of data points that I must fit to an inverse function like this:
y(x) = a/(x+b) + c
My problem is that least-squares fitting using this equation is extremely unstable and heavily dependent on initial guess. No matter how accurate parameters I start with, the algorithm often goes...
Is this normal? it doesn't seem correct.
The equation for the portion of circle with radius 1 unit in the 1st Quadrant is:
## y = f(x) = \sqrt{1-x^2} ## Domain is 0<x
But when I calculate f'(x) I also get
## f'(x) = \sqrt{1-x^2} ##
I thought inverse functions always reflect about y=x. Please...
Homework Statement
I will post a picture of the problem and then the second picture will be my work. The problems are the first two.
Homework EquationsThe Attempt at a Solution
I didn't know how to do this at first so I don't know if I am doing it correctly now. Also I don't know the correct...
Hello!
*Let $f$ be a strictly increasing continuous function on a closed interval $[a, b]$, let $c = f(a), d = f(b)$, and let $g:[c, d] → [a, b]$ be its inverse. Then $g$ is a strictly increasing continuous function on $[c, d]$.*
How can it be shown that $g$ is continuous at its endpoints $c$...
I am reading Manfred Stoll's book: Introduction to Real Analysis.
I need help with Stoll's proof of the Inverse Function Theorem (IFT) for real-valued functions of one real variable.
Stoll's statement of the IFT for Derivatives and its proof read as follows:
In the above proof we read:
"...
One semester I was asked to find the inverse of $\,f(x) \:=\:\dfrac{3x - 5}{2x+1}$
Later, I had to find the inverse of $\,f(x) \:=\:\dfrac{2x+7}{4x-3}$
It occurred to me that a general formula would a handy tool.
Especially since I planned to teach Mathematics and I might
be teaching this very...
Homework Statement
Find the formula of the inverse function of f(x)=300/(3+15e^.05x).
Homework Equations
f(x)=300/(3+15e^.05x)
The Attempt at a Solution
I'm definitely way off but I got .05y(5x)+ln100=lnx. What I did was multiple the denominator by the y(cross mltiplication)...
Hi, I have a relationship
$$P \cong \Bigg[\Big(K_1\rho^{\frac{5}{3}}\Big)^{-2}+ \Big(K_2\rho^{\frac{4}{3}}\Big)^{-2}\Bigg]^{-\frac{1}{2}}$$I need to find the inverse as $$\rho= \rho(P)$$.
I made a detailed calculation and came up to this
$$y^5+\Big(\frac{P}{K_2}\Big)^2 y+...
Consider the function g(x) represented by the table below:
x -6 -4 -2 0 2 4 6
g(x) -4 -2 4 0 6 -6 2
Complete the table of values for the INVERSE, g^{-1}(x), in the table below:
x -6 -4 -2 0 2 4 6
g^{-1}(x)
I just started Calculus 1, a summer quarter that's compressed and I'm having trouble understanding a theorem that state continuity of the inverse function. Within my textbook, it mentions "If f(x) is continuous on an interval I with range R, and if inverse f(x) exists, then the inverse f(x) is...
Homework Statement
I have a function f:M_{n×n} \to M_{n×n} / f(X) = X^2.
The questions
Is valid the inverse function theorem for the identity matrix? It talks about the Jacobian at the identity, but I have no idea how get a Jacobian of that function. Can I see the matrices as vectors and...