Question
Question: How do find the inverse of \[y=\dfrac{1}{x-3}\]?...
How do find the inverse of y=x−31?
Solution
In this question, we have to find the inverse of a function. This question is from the topic of pre-calculus. During solving this question, firstly we will see what are the steps to find the inverse of a function. After that, we will solve the question. The inverse of a function is the image of the function whose reflecting surface is the line y=x.
Complete answer:
Let us solve this question.
In this question, it is asked to find the inverse of y=x−31.
So, first let us know how to find the inverse of a function.
If a function y=f(x) is given and we have to find the inverse of that function where f(x) is a function of x.
Then, the first step will be replacing y by x and replacing x by y. This is done to make the rest processes easier.
After that, we will find the value of y in terms of x.
The value of y will be the inverse of function of x.
Using the above methods, we will solve the question.
As y=x−31, then we will replace y by x and x by y. We get,
x=y−31
Now, we will find the value of y with respect to x or in terms of x.
⇒1x=y−31
Doing cross multiplication, we can write the above equation as
⇒1y−3=x1
⇒y−3=x1
Carrying 3 to the right side of the equation, we get
⇒y=x1+3
Taking LCM and solving in the right side of the equation, we get
⇒y=x3x+1
Hence, we get that the inverse of y=x−31 is x3x+1.
Note: We should have a better knowledge in pre-calculus for solving this type of question. While finding the value of y in terms of x after replacing y by x and x by y, mistakes can be made. So, be aware of that. We have a method for verifying that if we are correct or not. Let us suppose we have a function f which is a function of x. And we have found that the inverse of f is f−1. Then, the relation f−1(f(x))=f(f−1(x))=x will always satisfy. For example, we can see in the equation y=x−31. We know that its inverse is x3x+1. Let us verify that.
Here, function is f(x)=x−31 and inverse of the function is f−1(x)=x3x+1.
So, f−1(f(x)) will be f−1(f(x))=f(x)3f(x)+1=(x−31)3(x−31)+1=x−31x−33+x−3=x−31x−3x which is equal to x. Hence, it is verified.