In
today’s article, we are going learn “how to find the inverse of a function”
Before learning how to figure out the inverse of a function, we need to first understand what is the inverse of a function, after that, we are going to learn their calculation procedure through each step with a practical example.
Techniques to Understand and Calculate the Inverse of a Function with Clear Explanations and Examples
If
f is a function then the set of ordered pairs that are generated by exchanging
the first and second coordinates of each ordered pair in f is referred to as
the inverse of f. It is indicated by f ⁻¹.
If
and only if a function f is injective and injective and surjective, then f has
an inverse.
Let
the function f: A → B exist. Another function, f - 1, is the inverse
of f: B →A.
Thus,
x
= f - ¹( y ) ⇔ y = f ( x )
Mastering Inverse Functions: Easy-to-Follow Techniques for Low-Competition Results
If
the function f is provided, then the inverse function f ⁻¹ may be obtained via algebraic
steps which require various stages.
Stage:
1
Express
the function y = f ( x ) expression.
Stage:
2
Determine
the problem in Stage: 1 for x in terms of y.
Stage:
3
In
the resultant equation in Stage: 2, replace x with f⁻¹ (y)
Stage:
4
Substitute
each y in the resulting output of Stage: 3 with x to obtain f ⁻¹ ( x ).
Stage:
5
Check
the answer by proving that
f
⁻¹ ( f ( x ) ) = x
Example of an inverse function:
Let
f: R → R be the function given by:
f(x)=
3x-8
Now
we have to find f ⁻¹ (x)
Stage:
1
Replace
x with y
Thus,
Y=3x-8
Stage:
2
Now
calculating the equation for x in terms of y
thus,
y+8=3x
x=y+8/3
Stage:
3
Now
replace x with f ⁻¹ (y)
So,
f
⁻¹ (y)= y+8/3
Stage:
4
In
this step, we have to replace y with x to acquire f ⁻¹(x)
Thus
we will get,
f
⁻¹(x)=x+8/3
Stage:
5
In
this stage, we have to verify our answer.
f ⁻¹ ( f ( x ) ) = x
Therefore,
f
⁻¹{f(x)}= f ⁻¹(x+8/3)
= f ⁻¹(x)=x+8/3
= (3x-8+8)/3
= 3x/3
= x
Thus,
it is proved
f ⁻¹ ( f ( x ) ) = x
