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.
This came up when I was trying to evaluate to a decimal value an arcsec(something), but on TI-83 there is no inverse secant button. So, I punched 1/(arccos(something)). Naturally, this came up as a domain error.
Then my instructor pointed out arcsesc is not 1/arccosx. Oops.
Thinking about...
Please refer to the attachment.
For part a)
so far I have:
$e^x = 1 + \frac{x}{1!} + ...+ \frac{x^n}{n!}$
So
$S^\frac{-1}{2}e^\frac{-1}{S} = S^\frac{-1}{2}(1 -\frac{1}{S} + \frac{1}{2!S^2} - \frac{1}{3!S^3} + \frac{1}{4!S^4} + ... - ...)$
I don't think my $S^\frac{-1}{2}$ on the outside is...
Homework Statement
Find (f−1)'(a).
f(x) = 5x^3 + 3x^2 + 5^x + 4, a = 4
Homework Equations
I'm not entirely sure but I assume I have to use d/dx(f-1) = 1/f '(f-1(x))
The Attempt at a Solution
So far I switched y and x. Found dx/dy to be 15y^2 + 6y + 5. Then I switched dx/dy to...
Homework Statement
Hey guys,
So I have the following permutations, which are a subgroup of S3:
σ_{1}=(1)(2)(3), σ_{5}=(1,2,3), σ_{6}=(1,3,2)
This is isomorphic to Z3, which can be written as {1,ω,ω^{2}}
Next, we have the basis for the subgroup of S3:
e_{i}=e_{1},e_{2},e_{3}
And we also have...
Hey guys!
Basically, I was wondering how to prove the following statement. I've seen it in the Hamermesh textbook without proof, so I wanted to know how you go about doing it.
Let's say you have a group element g_{1}, which has a corresponding inverse g_{1}^{-1}. Let's also define a linear...
Homework Statement
##{\Lambda_c}^b## is a Lorentz transformation and ##{\Lambda^c}_b## is its inverse, so ##{\Lambda_c}^b {\Lambda^c}_b## gives an identity matrix.
How can I write this, assuming it's possible, in terms of ##\delta##'s ?
Homework Equations
The Attempt at a...
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?
again, i need some help here guys.$\displaystyle\int\frac{3x-1}{2x^2+2x+3}dx$
=$\displaystyle\int\frac{3x-1}{2\left[\left(x+\frac{1}{2}\right)^2+\frac{5}{4}\right]}dx$
$\displaystyle a=\frac{\sqrt{5}}{2}$; $\displaystyle u=x+\frac{1}{2}$; $\displaystyle du=dx$; $\displaystyle x=u-\frac{1}{2}$...
Homework Statement
$${ { L } }^{ -1 }\{ \frac { s }{ { ({ s }^{ 2 }+1) }^{ 2 } } \} +{ { L } }^{ -1 }\{ \frac { 1 }{ { ({ s }^{ 2 }+1) }^{ 2 } } \}$$
Homework Equations
The Attempt at a Solution
I used ##{ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s...
Homework Statement
I need to prove that Q[√2,√3] = \{a+b√2+c√3+d√6: a, b, c, d \in Q\} is a field.
Homework Equations
The Attempt at a Solution
I proved all axioms except the existence of inverses for nonzero elements. My problem is that multiplication is quite hairy. I ended up...
Homework Statement
I have some problem finding the inverse laplace transformation of the function: \frac{s}{s^2+2s+5}
Homework Equations
http://math.fullerton.edu/mathews/c2003/laplacetransform/LaplaceTransformMod/Images/Table.12.2.jpg
The Attempt at a Solution
I tried to...
hey guys can you help solve this problem.
$\displaystyle\int\frac{dx}{\sqrt{2x-x^2}}$
i know that i have to change the integrand into this form $\displaystyle\frac{du}{\sqrt{a^2-u^2}}$ can you please show me how. thanks!
Homework Statement
I have that X is distributed with Gamma(a,b) and that Y = \frac{1}{X}. I found the pdf of Y to be \frac{1}{\Gamma(a)b^a} \left(\frac{1}{y}\right)^{a+1} e^{-1/yb} for y > 0. I need to use this to find the expected value.
Homework Equations
The gamma function is...
Homework Statement
The nonlinear, inherently unstable inverted pendulum is shown in
Figure 1.15. The goal is to maintain the pendulum angle θ(t) = 0
by using a feedback controller with a sensor (encoder or poten-
tiometer) for θ(t) and an actuator to produce an input force f (t).
The...
Homework Statement
I have two questions sin2x = 1/25 and this obviously becomes sinx= +-(1/5)
I also have cos2-1.5cosx-0.54 and cosx = (-3/10) and (9/5)
Now this is asking for me to solve for the x value in radians in the domain [0,2pi] and I have no idea how to solve these for exact values...
Homework Statement
I'm stuck trying to find out the inverse Laplace of F(s) to get y(t) (the solution for the differential equation):
Y(s) = 1 / [ (s-1)^2 + 1 ]^2
The Attempt at a Solution
I tried using a translation theorem and then apply the sine formula, but the denominator...
Does every FFT have \(i\) in it?
Given \(u_t = -(u_{xxx} + 6uu_x)\).
\(f'''(x) = \mathcal{F}^{-1}\left[(ik)^3\mathcal{F}(f(x))\right]\)
\(f'(x) = \mathcal{F}^{-1}\left[(ik)\mathcal{F}(f(x))\right]\)
The only equation I have used the pseudo-spectral method on was the NLS which is
\(u_t =...
Quoting from Wiki, bolded section mine:
What exactly is going on here with the data rate? Is it just the strength of the signal that is falling off as distance increases? If so, how does that reduce the data rates that can be used? If not, what limits the data rates?
I need some help understanding inverse functions, we've had a 4-page chapter covering the basics of inverse functions and I understand that.
But now we have suddenly gotten these task that I don't understand how to solve, I've read the part on inverses several times, but I still don't...
Within certain branches of analysis - both real and complex - the Inverse Tangent Integral (and its generalizations) can be quite useful. Similarly, it's much less well-known (= uglier? lol) cousin, the Inverse Sine Integral can be used to solve many problems.
To that end, this is not really a...
1. Homework Statement
A permanent electric dipole consisting of charges +q and -q
separated by the fixed distance s. Charge +Q is distance r from
the center of the dipole. We'll assume s << r.
a) Use the binomial approximation (1+x)-n ~= 1-nx if
x << 1 to show that the net force exerted...
q)give the converse ,the contrapositive and inverse of these conditional statements
a)if it rains today,then i will drive to work
b)if |x|=x then x>=0
c)if n is greater than 3,then n^2 is greater then 9
i know when you are dealing with limits you can take the inverse to fit the standard limit equations.
how about integrals? can u take the inverse for instance: integral(f(x)dx)
turn it into integral((1/f(x))dx)^-1) get the answer and then reverse it back?
when can u or can't you take...
1. Which of the following is a group?
To find the identity element, which in these problems is an ordered pair (e1, e2) of real numbers, solve the equation (a,b)*(e1, e2)=(a,b) for e1 and e2.
2. (a,b)*(c,d)=(ac-bd,ad+bc), on the set ℝxℝ with the origin deleted.
3. The question...
How does this work? I'm very confused about the phi is solved using inverse sin.
knowing: A=(c^{2}_{1}+c^{2}_{2})^{1/2} and c_{2}= Acos(\phi)
solve for \phi
which yields: \phi=sin^{-1}\frac{c_{2}}{(c^{2}_{1}+c^{2}_{2})^{1/2}}=tan^{-1}\frac{c_{2}}{c_{1}}
I'm not sure how we use the inverse...
Hi !
I've been thinking this problem a whole and I could not find an answer. I want to solve the following problem: suppose I have N mass particles with absolute coordinates \mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N . Besides, I have the following contraints: for all i=1,2,\ldots,N-1...
Homework Statement
Evaluate ∫(x^2-4)^(1/2) / x for x > 2
Homework Equations
The Attempt at a Solution
I was able to solve this problem via substitution, and my answer is: (x^2-4)^(1/2) - 2arcsec(x/2) + C. However, when I put the question into Wolfram Alpha, it gets this...
Homework Statement
The problem can be found in Jackson's book.
An infinitesimal Lorentz transform and its inverse can be written under the form ##x^{'\alpha}=(\eta ^{\alpha \beta}+\epsilon ^{\alpha \beta})x_{\beta}## and ##x^\alpha = (\eta ^{\alpha \beta}+\epsilon ^{'\alpha \beta})...
Homework Statement
I have to turn this homework in online... I just want someone to check my work
Convert from Cartesian coordinates to Polar coordinates
(-1,-sqrt(3))
if r > 0 and if r < 0.
Homework Equations
The Attempt at a Solution
if r > 0 then I believe the answer is...
I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question.
Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse...
I know the inverse of a function is found in two steps.
Isolate the independent variable and then switch the variables like this:
[y = x^{3} +1] = [x = \sqrt[3]{y - 1}]
Then switch the variables to get: y = \sqrt[3]{x-1}
However, when it comes to finding the derivative of the...
f(x) where x belongs to all real numbers
inverse: f-1(x), where x belongs to all real numbers
True or False:
The inverse of f(x+3) is f-1(x+3)
My ideas:
I think that it is false given that when you usually find the inverse of a function, you switch the x and y variables and solve for y again...
Homework Statement
Find the inverse of f(x)=ln(x^3-3x^2+3x-1)
Homework Equations
n/a
The Attempt at a Solution
y=ln(x^3-3x^2+3x-1)
x=ln(y^3-y^2+3y-1)
e^x=(y^3-y^2+3y-1)
i looked around for inverse of cubic functions and i found a monster of a formula...
Homework Statement
Find the inverse Laplace transform of e^(-3pi*s)/(s^2+2s+3).
Homework Equations
I know that you're supposed to factor out the e^(-3pi*s) and the other part becomes 1/(s+1)^2+2 but how do you get the answer? I'm confused.
The Attempt at a Solution
The answer is...
Homework Statement
Which of the following is the solution set of the equation
$$2\arccos(x)=\text{arccot}\left(\frac{2x^2-1}{2x\sqrt{1-x^2}}\right)$$
A)(0,1)
B)(-1,1)-{0}
C)(-1,0)
D)[-1,1]
Ans: A
Homework Equations
The Attempt at a Solution
I start by rewriting LHS in terms of ##\arctan##...
Homework Statement
The sum of the infinite terms of the series
\text{arccot}\left(1^2+\frac{3}{4}\right)+\text{arccot}\left(2^2+\frac{3}{4}\right)+\text{arccot}\left(3^2+\frac{3}{4}\right)+...
is equal to
A)arctan(1)
B)arctan(2)
C)arctan(3)
D)arctan(4)
Ans: B
Homework Equations
The Attempt at...
Homework Statement
Find the Inverse Laplace Transform of \frac{1}{(s^{2} + 1)^{2}}
Homework Equations
The Attempt at a Solution
I tried using partial fractions but it didn't work. It looks like a cosine transform, but I don't know what else to do. Help please :(
Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :)
Part B
2) Given the functions f(X) = 7x^2 - 5x and g(x) = 2x - 3 determine and simplify the following:
a) (f-g)(x)
My Answer: 7x^2 - 7x - 3
b) (f - g)(2)
My...
Homework Statement
Find the inverse Laplace transform of F(s)=(2s-3)/(s^2-4).
Homework Equations
I don't want to find the answer by looking at the Table.
F(s)=2s/(s^2-4)-3/(s^2-4)
The Attempt at a Solution
The answer is f(t)=2 cosh 2t - (3/2) sinh 2t.
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...
Hello,
I have the following matrix of matrices
\mathbf{H}=\begin{array}{cc}\mathbf{A}&\mathbf{B}\\\mathbf{B}^H&\mathbf{A}\end{array}
where each element is a square matrix, A is a diagonal matrix of real numbers, whereas B is not (necessarily), and the superscript H means conjugate transpose...
Write the inverse, converse, and contrapositive of the following statement:
upside down A x E R, if (x + 2) (x - 3) > 0, then x < -2 or x > 3
Indicate which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a conterexample for each that...
Let f(x) = k\ log_2 x
(a) Given that f^{-1}(1)=8, find the value of k
to get f^{-1}(x) exchange x and y
x=log_2 y^k
then convert to exponential form
2^x=y^k then 2^{\frac{x}{k}} = y
so for f^{-1}(1) = 2^{\frac{1}{k}}= 8=2^3 then \frac{1}{k}=3 so k=\frac{1}{3}
(b) find...