Find the inverse of each function f(x) given, then prove (by composition) your inverse function is correct. Note the domain of f is all real numbers.

Question

Find the inverse of the function f(x) given, then prove (by composition) your inverse function is correct. Note the domain of f is all real numbers.

Answer 1

Let f(x)=(2x+1)^1/3 be y=(2x+1)^1/3

Switch x and y to get the inverse function:

x=(2y+1)^1/3

Solve for y:

x³=2y+1

2y=x³-1

y=(x³-1)/2

To prove by composition, let the above inverse function be g(x)=(x³-1)/2

Find f(g(x)) and g(f(x))

f(g(x))=((2(x³-1)/2)+1)^1/3=x

g(f(x))=(((2x+1)^1/3)³-1)/2=x

Since f(g(x))=g(f(x))=x, we can conclude that the inverse is correct.

Have a good night! :)

Answer 2

First rewrite the function in exponential form

f(x) = (2x + 1)^1/3

let f(x) be y and switch x and y to get the inverse function.

y = (2x + 1)^1/3

x = (2y + 1)^1/3

And solve for y in terms of x

x³ = 2 y + 1

2 y = x³ - 1

y = (x³ - 1)/2