Domain of a relation

The domain of any function is known as the set of values that are allowed to plug into the function. There are different procedures to find domains in different types of functions. The domain of inverse function is the same as the range of the original function. For example, if the output of the function f is the input to f-1, then the range of the function f is also the domain of f-1 and vice versa.

Suppose there exists no inverse function for a function it is possible to create a function with inverse by specifying some boundaries in the original function . The new function will have an inverse function on a limited domain. Let’s take an example, the inverse of f(x)=x

It will be f-1(x)=x2, since the square is the inverse of square root on the domain(0,), and it will also be the range of f(x)=x.

Suppose we solve this problem from the other perspective, the function f(x)=x2. Now, if we construct an inverse to this function, this will become difficult because there are mainly two corresponding inputs for every function’s output. For instance, if we have an output of 9 of a quadratic equation with inputs 3 and -3. But output from a function is also an input to its inverse. 

Therefore, if the inverse input’s value is more than one of the inverse outputs, then the inverse is not a function. In other words, to have an inverse, the function is not an one to one function because it fails the horizontal line test and thus does not have an inverse function. The function must be one-to-one for a function to have an inverse. 

In other cases, if the function turns out not to be a one-on-one, you can still restrict it to the part of its domain where it is one to one function. 

Solved examples

Example 1: Find the domain and range of the inverse function of the given function f(x)=x3

Solution: finding inverse function of the given function 

y=x3

x=y13

Therefore, f-1(x)=3x

Now, look at the graph of f-1(x)=3x

We can see that both f-1(x) and x can be an element of a real number. Hence domain and range of f-1(x)=R.

Example 2: Find the domain and range of the following function. Also, find the inverse function and list its domain and range. f(x)=1x+2

Solution: the denominator of this function is not equal to zero, thus x+20. This means that x-2, therefore, the domain is all real numbers except -2.

Domain of f : (-,-2) U (-2,)

As we know, the domain of the function and the range of the inverse function is always the same. This is the outcome of, when changing the function to its inverse, we also switch the outputs and the inputs. If the domain of f is real numbers except -2, then the range of f-1 is the same. 

Range of f-1: (-,-2) U (-2,)

If you want to find the range of the original function f(x)=1x+2 , First, you need to find its inverse function. Because the range of f will be as same as the domain of f-1. To find the inverse follow these steps:

f(x)=1x+2

f=1x+2

f(x+2)=1

xf+2f=1

xf=1-2f

x=1-2ff

f-1(x)=1-2Xx

This is the inverse of the function, and the inverse function is rational, just like the original function. You can directly find the domain by stating that the denominator can never be equal to zero. Therefore, x0and all domain of f-1 is a real number except 0.

Domain of f-1: (-,0) U (0,)

As we all know, if this is the domain of f-1 , it is also the range of f. 

Range of f: (-,0) U (0,)

Example 3: find the domain and range of the following function. Also, find the inverse function and list the range and domain of inverse function.(a) f(x)=-21-x (b) f(x)=-2-xx+1 

(a) Solution: the denominator of this function is not equal to zero, thus 1-x0. which means that 1x, 

Domain of f : (-,1) U (1,)

Range of f-1: (-,0) U (0,)

f(x)=-21-x

f=-21-x

(1-x)f=-2

f-fx=-2

f+2=fx

x=f+2f

f-1(x)=x+2x

x0

This is the inverse of the function, and the inverse function is rational, just like the original function. You can directly find the domain by stating that the denominator can never be equal to zero. Therefore, x0and all domain of f-1 is a real number except 0.

Domain of f-1: (-,0) U (0,)

As we all know, if this is the domain of f-1, it is also the range of f. 

Range of f: (-,1) U (1,)

(b) Solution: the denominator of this function is not equal to zero, thus 1+x0. which means that -1x, 

Domain of f : (-,-1) U (-1,)

Range of f-1: (-,-1) U (-1,)

f(x)=2-xx+1

f(x+1)=2-x

f+fx=2-x

xf+x=2-f

x(f+1)=2-f

x=2-ff+1

f-1(x)=2-xx+1

x+10

x-1

This is the inverse of the function, and the inverse function is rational, just like the original function. You can directly find the domain by stating that the denominator can never be equal to zero. Therefore, x0and all domains of f-1 are a real number except 0.

Domain of f-1: (-,-1) U (-1,)

As we all know, if this is the domain of f-1, it is also the range of f. 

Range of f: (-,-1) U (-1,)

f(x)=2-xx+1 is a special case where the inverse of the function and the function are similar. 

Conclusion 

When the original function is a quadratic equation, by observing the inequalities with the function, you can tell what will be the function’s domain and give you information about which sign to put before the square root in the inverse function. The algebraic determination of the formula for an inverse function is y=f(x); you are to switch the roles of x and y, for example, x=f(y), and then further simplify the expression for y, and you will get y=f-1(x). If you switch the roles of x and y, it also affects the role of domain and range of a function f(x).