What is the relationship between the domains and ranges of a function and its inverse?

Answer 1

The domain of a function, f(x), is a set of real numbers (call them values of x) which corresponds to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range ( y- value).

So the range is the set of real numbers that are values of the function.

An inverse of a function f(x) is denoted by f-1(x) where -1 is NOT an exponent. The notation f-1 does not mean 1/f (so it looks like a neg 1 exponent but it is not. Math people know to read this as the inverse function).

Any function that passes the horizontal line test (which intersects the graph of the function only once) has an inverse, also it is a one-to-one function. Any one-to-one function has a graph that passes the horizontal line test.

A one-to one function is a function in which not two different pairs have the same second component. For this kind of functions (one-to-one functions), the domain becomes the range for the inverse and vv. It means that if a point (x, y) is on the graph of f, then the point (y, x) is on the graph of f-1.

Ex: y or f(x) = x2

(the domain is the set of all real numbers. you can square positives, negatives, fractions etc. the range is only all reals greater than or equal to zero).

The graph of f(x) = x2 does not pass the horizontal test, because it intersects the graph at two points, let's say (-3, 9) and (3, 9). Inverse functions have ordered pairs with the coordinates reversed. If we interchange x- and y-coordinates then we obtain (9, -3) and (9, 3) but these ordered pairs do not define a function. Thus this function does not have an inverse. But if we restrict the domain, for example the set of all positive numbers including zero, then we allow it to have one, and this inverse function f-1 is a reflection of the graph of f about the line y = x, where f(x) = x2 and its domain is {x| x ≥ 0}.

The inverse of the above function is the square root of x. which I will abbreviate as sq rt

the inverse function becomes f-1(x) = √x

(in other words, f you limit yourself to real numbers, you cannot use any negatives in place of x for this inverse function. So the domain of the inverse is all reals > or = 0.

If the inverse is to be a function you cannot have any answers which are negative. the relation would not pass the vertical line test. so the range is also only reals > or = zero).