Onto and Inverse Functions

The inverse function is denoted as f-1 since it only exists when f inverse is one-one or onto. F inverse is not the reciprocal of this function’s or its reciprocal function’s composition if it gives the dominant value x.

Moreover, the inverse of f inverse in the given function has a domain of Y that belongs to Y. It is related to a distinct element X belonging to X in the co-domain set, and this kind of function is also called the onto function or Jackson function. When the inverse of the function is both injective and surjective, it is called a bijective function.

Onto function or surjection:

A function  f: P→Q is called an onto function if every element of Q has at least one preimage in P. In other words we can also define,  f: P→Q is called a function of P onto Q, if the image set of P under F(i.e, range of function) is equal to its co-domain Q ie, {f(x): x∈P}=Q.

So,   f: P→Q  is a surjection (or onto mapping) then corresponding to every element y∈ Q there exist at least one element x∈P such that f(x)=y

It should be noted that, if P and Q are two finite series then the function  

f: P→Q is a surjection , when n(P)>= n(Q). 

Ex:

 f(x)= 2x onto?

This is an onto function as it has at least one preimage for every output. 

If f: A→B is one-to-one, there is at least one element “a” in the domain such that f(a) = b, that is, the function maps one or more elements of A to the same element of B for every element “b” in the co-domain B.

Inverse function

An inverse function is one that can be reversed. If a function y can be written with respect to x and x can be written with respect to y, it is called an inverse function. 

In such a function, say:

f(x) = 2x+3 = y

This function will also work as

x = (y – 3)/2

In this case, the inverse function will be x = f-1(y) 

So, f-1(y) = (y-3)/2

In other words, an inverse function can be reversed into another function. If an inverse function takes you from x to y, it can also lead you from y to x.

Example:

f(3) = 2×3 +4 = 10

f-1 (10) = (10-4)/2 = 3

Inverse function graph

To draw an inverse function on a graph, you have to draw a line y = x. Two lines will be drawn to show an inverse function that are symmetric. One line is denoted as f(x) and the other as g(x). The two lines of the inverse curve are symmetric on the line y = x.

Example:

Take an inverse function, say,

f(x) = 2x – 3 = y

Now, we have to calculate the inverse function of this function, that is,

f-1 (x) =( y+3)/2

The function is an inverse function – to check it, put 4 as the value of x.

f(4) = 2*4 – 3 = 5 = y

You will get the value of y. Now substitute the value of y.

 f-1 (5) = (5+3)/2 = 4 = x

You will get the value of x!

Steps for inverse function

As we know, a function whose output can be used to find the original value is an inverse function. This is the process to calculate the inverse of a function.

• The first step is to replace f(x) with y to make the steps easier.
• The second step is to solve the equation and find the value of y. Follow the equation rules carefully to avoid mistakes in this step.
• We have both functions now: f(x) and f-1 (x).
• Now it’s time to verify the irreversibility. Put the value of x in f(x) and find the equation with y.
• Now put the value of y, and you will get the value of x.

For example:

y = 2x – 4=f(x)

Now put y in the place of f(x)

y = 2x – 4

Now find the inverse function

x = (y + 4 )/ 2

⇒ f-1 (x) = (Y + 4) / 2

Now verify the irreversibility of the function

Put 2 as the value of x

f (2) = 2 * 2 – 4 = 0 = y

Now put the value of y in inverse

f-1 (0) = (0 + 4) / 2 

⇒ 2 = x

Follow these above steps and try to find the inverse function by yourself now!

Conclusion

An inverse function is a function that can be reversed. Simply, if a function f with respect to x equals to y or f(x) = y, then f-1(y) = x is true, where f-1(y) is the inverse of f(x). You can get the original value from the output of the inverse function. The two lines of the inverse curve are symmetric on the x = y line. There are mainly four types of inverse functions: trigonometric functions, rational functions, hyperbolic functions, and log functions.

When all the elements in a co-domain are mapped from a domain, that is, the co-domain is equal to the range, it is a surjective function.