Is the domain of a function equal to the range of its inverse?

Question

This is true for simple functions like f(5)=11
In this case the inverse will be f^-1(11)=5. So the domain used to be 5 but after inversion the domain became 11. The range used to be 11 but after inversion the range changed to 5. The domain of the function (5) is equal to the range of its inverse (5).

So, my question is, does this rule always apply?
What if I have f(x)=2x+1? The inverse is f^-1(x)=x-1/2 but it does not seem that the domain of the function is equal to the range of its inverse!

If x was equal to 2 then the domain of the function will be 2 but the range of the inverse will be 1/2 which is obviously unequal...

Idk, I'm kind of confused about this...help

Thanks

Answer 1

f(x)= 2x+1

its inverse is y= (x-1)/2 or 0.5x-0.5



If x was equal to 2 , f(2)=2(2)+1 = 5



use 5 as input for inverse, you should get the input from function which is 2

lets try



y=(5-1)/2 = 4/2 = 2



Your question is asking does this rule always apply.

Yes. it does! Function's domain is equal to range of inverse and Inverse's domain is equal to range of function

You just have to make sure that the function even have an inverse and be sure to solve the inverse function correctly !



Oh and.. how to find out if the function have inverse or not?

The function must pass the horizontal line test and the vertical line test.

Answer 2

What if I have f(x)=2x+1? The inverse is f^-1(x)=x-1/2...

actualy the inverse f^-1(x) = x/2 - 1/2



and we restrict the domain of f(x) to the integers {1, 2, 3}

f(1) = 1*2+1 = 3

f(2) = 2*2+1 = 5

f(3) = 3*2+1 = 7



The range of f(x) = {3, 5 and 7}

{3,5,7} are the domain of f^-1(x)

f^-1(3) =3/2 - 1/2 = 1

f^-1(5) =5/2 - 1/2 = 2

f^-1(7) =7/2 - 1/2 = 3



is this always the case...



f(x) = x^2 f^-1(x) = sqrt (x)

the domain of f(x) is all real numbers

the range of x is positive real numbers x^2 is always >=0



The domain of f^-1(x) is positive real numbers... so far so good.

The range of f^-1(x) is positive real numbers.



f(x) is not a 1 to 1 function. When we find f^-1 (x) we need to restrict the range such that for any x there is only one f^-1(x).

Answer 3

Most functions don't have inverses. 1-1 functions have inverses. Then what you say is true.



Be a bit careful, though. For example when you say f(5) = 11 you are saying that the function f assigns the number 11 to the number 5. It does NOT mean that the domain of f is 5. For example, when f(x) = 2x + 1, then f(5) = 11 but the domain of f is the set of all real numbers.

Answer 4

Yes, it's true



Domain of f(x) = R = Range of f^-1(x)