Question

Find the inverse of f(x)= $(4-(x-7{)}^3{)}^{1/5}$

• A. 7+ (4-x⁵)^1/3
• B. (4-(x-7)³)^1/5
• C. 4-(x-4)²
• D. 7+ (4+x⁵)^1/3

Answer 1

The correct option is A

7+ (4-x⁵)^1/3

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

y${}^5$ = 4 - (x-7)${}^3$

(x-7)${}^3$= 4- y${}^5$

Taking cube root on both sides

x-7 = (4 - y${}^5$)${}^{1/3}$

x = 7 + (4 - y${}^5$)${}^{1/3}$

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 - x${}^5$)${}^{1/3}$. 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)