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.

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

To find the inverse function of y = cbrt(2x + 1), we swap x and y, then solve for x. We have:

x = cbrt(2y + 1)

x³ = 2y + 1

2y = x³ - 1

y = 0.5(x³ - 1)

Therefore, we have f^-1(x) = 0.5(x³ - 1). To prove that the inverse function is correct, we must show that f(f^-1(x)) = x. We have:

f(f^-1(x)) = cbrt(2(0.5(x³ - 1) + 1)) = cbrt(x³ - 1 + 1) = cbrt(x³) = x.

Therefore, our inverse function is correct.