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Inverse of a Function

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The Inverse of a Function, a selection of answers from the Dr. Math archives.

What is an inverse?

Inverse Statements
Say there are 3 + 4 times as many girls as boys. Is this the same as saying there are -3 + 1/4 as many boys as girls?

Inverting Functions
To find the inverse of a function y=f(x), do I interchange the variables x and y, or do I solve for x in terms of y?

Inverse Functions in Real Life
I would like to know if you have a good example of how inverse functions would be used in real life.

Is the Function Invertible?
For the following functions f(x) decide if the function is invertible as a function from R to R...

Invertible Functions
Consider as a function from R -> R (Real) and say whether the function is invertible: h(x) = (sgn x)* sqrt(abs(x)) where sgn is +1 if x is positive, -1 if x is negative, and 0 if x is 0.

Finding an Inverse Function
What is the inverse function of f(x) = 1 - 2x? What is the general method for finding an inverse function?

Is the Inverse a Function?
How do I find the inverses and determine whether the inverses are functions?

Unraveling an Inverse Function
I got this question wrong: y = (-5x - 2) / (-x + 1).

Function Its Own Inverse
Show that f(x) = X/X-1 is its own inverse.

Inverses of Trigonometric Functions
How can the functions cosine and sine have inverses of arcsin and arccos respectively?

Inverse Functions: Arcsec(x)
If y = arcsec(x) then what does x equal? How is this problem related to inverse functions?

Inverse of a Function vs. Inverse Proportionality
Finding the inverse of a function and graphing yields a graph that has been reflected in the line y = x relative to the function. Inverse proportionality, however, yields a reciprocal relation graphically. Why do these two things have similar names yet mean different things?

Inverse Functions One-Way Only
Are there two functions, f and g, such that (f of g)(x) = x but (g of f)(x) does not equal x?

Domain and Range of an Inverse Function
Why is the domain of a function equal to the range of the inverse function?

Inverse Functions - Which Statement is True?
Does (f o g)^(-1) equal f^(-1) o g^(-1) or g^(-1) o f^(-1)? How can you prove it?

Inverse of a Multivariate Function
Let f:NxN -> N such that f(x,y) = 2^x(2y + 1) - 1 for all natural numbers x, y. Let the inverse of f, g be given by g:N -> NxN. Find the inverse of the function g.

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