Question

Match each function to its inverse. Check your work by confirming that the compositions f(g(x)) and g(f(x)) both equal x. The functions are f(x)=x^3+5, f(x)=2/x-3, f(x)=4x-1. The inverses are f^-1(x)=x+1/4, f^-1(x)=3/x-5, f^-1(x)=2/x+3.

Answer 1

The correct inverses are:

- f(x) = x^3 + 5 ⇔ f⁻¹(x) = (x − 5)^(1/3).
Check:
f(f⁻¹(x)) = ( (x − 5)^(1/3) )^3 + 5 = x − 5 + 5 = x.
f⁻¹(f(x)) = (x^3 + 5 − 5)^(1/3) = (x^3)^(1/3) = x (for real x).

- f(x) = 2/x − 3 ⇔ f⁻¹(x) = 2/(x + 3).
Check (watch domains: x ≠ 0 for f, x ≠ −3 for f⁻¹):
f(f⁻¹(x)) = 2 / (2/(x+3)) − 3 = x + 3 − 3 = x.
f⁻¹(f(x)) = 2 / ( (2/x − 3) + 3 ) = 2 / (2/x) = x.

- f(x) = 4x − 1 ⇔ f⁻¹(x) = (x + 1)/4.
Check:
f(f⁻¹(x)) = 4((x + 1)/4) − 1 = x + 1 − 1 = x.
f⁻¹(f(x)) = (4x − 1 + 1)/4 = x.

(If the inverse expressions you were given were written without grouping symbols, they may have been ambiguous; the inverses above are the unambiguous correct ones.)