# 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.

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1. ### State the domain and range for a given function

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)\}...
2. ### B A function and its inverse

I could only verify this for a few elementary functions. Does a proof exist? Does it go beyond the realms of high school mathematics? Many thanks.
3. ### I Local inverse of non bijective functions

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!!
4. ### To find the range of a given ##\sin## function

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 ...
5. ### Interchanging x and y for inverse function

For this, Why are we allowed to interchange x and y? Is it because the equation will still be true? Many thanks!
6. ### I Can an inverse function of a special cubic function be found?

Like this one: ##f(x)=3x^3 -18x^2 +36x##.Is there any way to find the inverse of this function without using the general solution to cubic equation?
7. ### I Is the solution to this problem as trivial as I think?

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...
8. ### If f(x)=(e^x+e^-x)/2, what is the inverse function?

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

10. ### MHB -apc.1.1.5 inverse 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 $g'(1)?$ 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...
11. ### Having trouble finding the derivative of an inverse function

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.]
12. ### I want to see the plot of an inverse function -- can I use a TI84?

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...

48. ### How to find inverse function

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)
49. ### MHB Continuity of the Inverse Function

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...
50. ### Inverse function theorem over matrices

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...