What is Inverse functions: Definition and 103 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.
Consider the case of a real function f for which f inverse exists.
1) We we are not used to having the y-axis (vertical axis) to denote the independent variable which it does in x=f-1(y). We rotate the system through positive 90 degree and reflect about the vertical to change the sense of the...
1. a.
fg(x)=2(1/2(x-1))+1
fg(x)=2(x/2-1/2)+1
fg(x)=x-1+1
fg(x)=x
gf(x)=1/2((2x+1)-1)
gf(x)=1/2(2x+1-1)
gf(x)=x+1/2-1/2
gf(x)=x
The functions functions f(x) and g(x) are inverses of each other. This can be demonstarted by
f(x)=2x+1
y=2x+1
x=2y+1
x-1=2y
(x-1)/2=y
Thus, y=1/2(x-1) = g(x)
And...
f-1(f(A)) = A and f-1(f(B)) = B so options (a) and (c) are wrong.
For (b), I get A ⊆ A
For (d), I get B ⊆ B
For (e), I get A ⊆ A
So there are three correct statements? Thanks
ok I have been trying to cut and paste in packages and code to get a simple inverse function to plot
but nutin shows up and get error message.
if possible I would like no grid but an xy axis with tick only where the graph goes thru the axis
and of course a dashed line of x=y
some of the...
Write
$\cot^2(x)-\csc^2(x)$
In terms of sine and cosine and simplify
So then
$\dfrac{\cos ^2(x)}{\sin^2(x)}
-\dfrac{1}{\sin^2(x)}
=\dfrac{\cos^2(x)-1}{\sin^2(x)}
=\dfrac{\sin^2(x)}{\sin^2(x)}=1$
Really this shrank to 1
Ok did these on cell so...
I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...
I am focused on Chapter 4: Limits and Continuity ... ...
I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...
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
Find the smallest value of b so that the function f(x) = x^3 + 9x^2 + bx + 8 is invertible.
Homework EquationsThe Attempt at a Solution
I know that the function has to be only increasing/decreasing, and I think it is needed to find the derivative of the function. I do...
One of our homework problem asks:
If f is a one-to-one function such that f(-3)=5 , find x given that f^-1 (5)=3x-1.
Here's how I attempted to solve the problem:
-3=3x-1
3x=-2
x=-2/3
Is this the correct way to solve the problem?
If we have a relation, ##R##, and it's inverse, ##R^{-1}## they behave such that a point on ##R##, say (a,b), corresponds to the point (b,a) on ##R^{-1}## This is a reflections across the line y=x.
This relation does not mean that ##R^{-1}## is a function. For example,
Let ##R## be...
Temperatures can be converted from Fahrenheit to Celsius using the
function f(x) = 5
/9
(x − 32).
(a) Calculate f(59).
(b) Find f
−1
(x), and verify that f
−1
(f(59)) = 59.
(c) Let K be the set {x : f(x) = x}. Find all elements of K and list K
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
Homework Statement
Suppose that f has an inverse and f(-4)=2, f '(-4)=2/5. If G= (1/f-1) what is g '(2) ?
If it helps the answer is (-5/32)
Homework Equations
[/B]
f-1'(b)=1/(f')(a)
The Attempt at a Solution
Im not really sure how to start this problem. I am familiar with how to use the...
I'm working through the problems in the Mooculus textbook as revision for Calculus I & there seems to be something wrong with how I'm manipulating the function to find its inverse in the following example.
Homework Statement
The height in meters of a person off the ground as they ride a Ferris...
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...
Homework Statement
If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x).
Homework Equations
(None I believe.)
The Attempt at a Solution
I know that for a function to be invertible, it must be both one-one and onto...
Homework Statement
If we shift a curve to the left, what happens to its reflection in the line y = x? In view of this geometric principle, find an expression for the inverse of g(x) = f(x + c) where f is a one-to-one function.
Homework EquationsThe Attempt at a Solution
Initially I did this...
Homework Statement
Verify that f has an inverse <- prof told us not to worry about this. Then use the function f and the given real number a to find (f^-1)'(a).
f(x) = Cos(2x), 0<=x<=pi/2 where a=1
Homework Equations
1/(f'(g(x))) where g(x)=f^-1(a)
d/dx(cos(2x)) = -2sin(2x)
The Attempt at a...
I've always been having trouble with the domain and range of inverse trigonometric functions. For example, let's start with an easy one: $\sin^{-1}\left({x}\right)$
Process: First, I draw out the function of $\sin\left({x}\right)$. Then I look at its range and attempt to restrict it so that it...
Hey guys,
I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help.
Question:
Alright, I'm having quite a bit of trouble with these. So here it goes:
For the first one, I did the 3-step procedure to finding the...
In the Math Challenge Forum it has been requested fo compute the series...
$\displaystyle S = \sum_{n=1}^{\infty} \tan^{-1}\ \frac{\sqrt{3}}{n^{2} + n + 3}\ (1)$
... and that has been performed using the general identity...
$\displaystyle \sum_{n=1}^{\infty} \tan^{-1}\ \frac{c}{n^{2} + n +...
Can someone explain why the answer is D
a < 0 because it finishes downwards
e < O because the y-intercept is in the negatives.
b, & d = zero (but i don't get this)
c is supposedly > 0 (nor do i get this)
According to the solutions the graph is an even function, and symmetrical about the...
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...
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...
In www.mathhelpforum.com and interesting question has been proposed by the user misiazeska the 05 20 2013...How to find the inverse of this function?...$$y= 5\ x^{3} - x^{5}$$ ... and the unanimous answer has been '... it doesn't exist any closed formula to find x as function of y...'. In my...
I was just wondering if inverse functions only apply to one-to-one functions?(Or a function who's domain has been restricted to act as a one-to-one function). Thanks.
f(x)=a+4/(x-a)
f-1(x)=(a^2-ax-4)/(a-x)
Which of the following is true?
The function is the opposite of its own inverse for any value of a.
The function is its own inverse for positive values of a only.
The function is the reciprocal of its own inverse for positive values of a only.
The...
Homework Statement
5. (a) The functions f and g are defined by
f : x|→ 2x + ln3 (x is a real number)
g : x|→ e^3x (x is a real number)
i) Find f^-1(x) and g^-1(x) and state their domain of definition
ii) Show that f^-1 f = f f^-1 = x (x is a real number)
iii) Find the composite function...
Homework Statement
1. Show that the function f(x) = x5 -x3 +2x is invertible. Compute the derivative of f-1
at 2.
The Attempt at a Solution
To find f-1 I switched x and y which gave me x = y5 - y3 + 2y
this is where i got stuck because I am not sure how to solve for y after that...
Here is the question:
Here is a link to the question:
A math function question please help? - Yahoo! Answers
I have posted a link there to this topic so that the OP may find my response.
Hi I'm reading a book called Calculus lifesaver, and in the book they state that the inverse of a function f(x)= x3 is the same as f-1(x)=3√x and is the same as f-1(y)=3√y
So I did a test, with a simpler function and I can't see how this is true
If I have a function f(x)= 2*x
Then...
Hello,
let's suppose I have two functions \phi:U\rightarrow V, and T:V\rightarrow V that are both diffeomorphisms having inverse.
Furthermore T is linear.
I consider the function f(u) = (\phi^{-1}\circ T \circ \phi)(u), where \circ is the composition of functions.
Since T is linear, we...
Homework Statement
Find the derivative of:
1. f(x)=arccos(5x^3)
2. f(x)=∫cos(5x)sin(5t)dt when the integral is from 0 to x
Homework Equations
Chain rule, dy/dx=dy/du*du/dx
The Attempt at a Solution
For the first one, I can just take 5x^3 as u and then apply the chain rule...
Homework Statement :
Define f: ℝ→ℝ by f(x)=x^2. Find f^-1(T) for each of the following:
(a) T = {9}
(b) T = [4,9)
(c) T = [-4,9]
The attempt at a solution:
So, the inverse of f should be f^-1(T)=+/-√(x). Therefor:
(a) f^-1(9)= +/- 3
(b) f^-1(4)= +/- 2, f^-1(5)= +/- √(5), f^-1(6)= +/- √(6)...
Hi. I am reviewing for an exam, and this is a topic that I did not go through very thoroughly.
I understand how to calculate the derivative of an inverse function when I am given a point, as I simply use the equation (1/f((f^-1))'. So if I am given the equation y=x^3, for example, and am asked...
I have a graph f(x) = 3(x + 1)^2 - 12 , I have sketched this graph (Not shown) hand it is a parabola with a y- intercept at - 9. the vertex being - 12.
The image set is a closed interval {- 12, infinity} Sorry no square brackets and no sign for infinity.
I am asked to explain why the function...
Ok,
I understand an inverse function sends a variable in the range to the corresponding value in the domain, but am not sure if what I'm thinking is correct... : For example:
Let A be the set
A = \{1,2,3,7,8\} ; B = \{4,5,6\} and the function f map A to B s.t
f(1) = 4
f(2) =...
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...
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
Let f : A → B be a function and let Γ ⊂ B × B be an equivalence relation on B. Prove that the set (f × f)^-1 (Γ) ⊂ A × A (this can be described as {(a, a′) ∈ A × A|(f(a), f(a′)) ∈ Γ}) is an equivalence relation on A.Homework Equations
The Attempt at a Solution
Let...
Homework Statement
For exercises 49 and 50 let f(x) = (ax + b)/(cx + d)
50. Determine the constants a, b, c, d for which f = f-1
Homework Equations
I found in question 49 when they asked to find f-1 that:
f-1 = (dx - b)/(a - cx)
This was also the answer at the back of the book but...
Homework Statement
2 problems, i solved both of them but I am not 100 % I am right
Find all points on the curve y=x-2cosx where the tangent line to the curve is parallel to the line y=x and write an equation of the tangent line at such point
Let f^-1 be the inverse of the one -to- one...
Problem: If g(x) = 3 + x + e^x find Inverse of g at 4
My work:
4 = 3 + x + e^x
1 = x + e^x
This is where I stop... I can look at it and see that x = 0
But I don't know how to find the solution algebraically...
Homework Statement
let f(x)=(4t^3+4t)dt(between 2 and x)
if g(x) = f^(-1)(x), then g'(0)=?
Homework Equations
The Attempt at a Solution
f'(x) = 4x^3+4x
annd i already don't know where to go from here.. help?
Homework Statement
f(x) = 3x^2-6
We are asked to solve for the inverse function of the above function.
Homework Equations
The Attempt at a Solution
y=3x^2-6
x=3y^2-6
\frac{3y^2=x+6}{3}
y^2 = \frac{x+6}{3}
\sqrt{y^2}= \frac{\sqrt{x+6}}{3}
y=...
Hello guys :smile:
Given that: f(x) = e2x and g(x) = (2x-1)
Find: (g ° f)-1(x)
So, what I did first was to put f into g:
2 x e2x - 1 = y
2 x e2x = y + 1
e2x = y + 1 / 2
(ex)2 = y + 1 / 2
ln (y+1/2) = x
Is that ok?
Thanks,
Peter G.