Answer: Option A
:
A
Conventional method
Replacing f(x) with y, the given function is
y= (4-(x-7)³)^1/5
To find the inverse function, we need to write an equation for x in terms of y
Raising both sides to the power 5, we get
y⁵ = 4 - (x-7)³
(x-7)³ = 4- y⁵
Taking cube root on both sides
x-7 = (4 - y⁵)^1/3
x = 7 + (4 - y⁵)^1/3
We have an expression for x interms of y. We get f^-1(x) if we replace y with x in the above equation
Hence, f^-1(x) = 7 + (4 - x⁵)^1/3. Answer option (a)
Shortcut
What is the basic definition of an inverse function?
An Inverse function is one which can be expressed as f(y) = x when f(x) = y
Using, this very definition, we can use a common sense approach to arrive at the answer In no time at all
1) Substitute a suitable value for x (it can be any value). in this case, for our convenience we will take x=7. Then at x=7, f(x)= 4^1//5
2) Now use the definition of inverse function. Put x=4^1/5 in each of the answer options and see where you get f(x)=7. You are just exchanging "x” and "y” values!
This is the concept of inverse function after all!
Only option (a) gives 7 at x=4^1/5
All other options can be eliminated as for x=4^1/5, we do not get 7. Answer is option (a)