Question

What is the inverse of the function f(x)=x ^6? Write your answer in the form a(bx+c) ^2+d, where a, b, c, and d are constants. Enter the domain of the inverse in the form x≤v.

Answer 1

The inverse function of f(x) = x^6 is y = x^(1/6), which cannot be expressed in the form a(bx+c)^2+d as requested. The domain of the inverse function is x ≥ 0.

Step-by-step explanation:

To find the inverse of the function f(x) = x6, we first replace f(x) with y to get y = x6. Then, we swap x and y to get x = y6 and solve for y. To do this, we take the sixth root of both sides so that y = x1/6 or y = (x)1/6. However, expressing y in the form a(bx+c)2+d isn't possible because the sixth root function isn't equivalent to a function raised to the second power.

In terms of the domain, since the original function has the domain of all real numbers and the function is even (its graph is symmetrical about the y-axis), the domain restriction will be x ≥ 0 for the inverse function because we only consider the principal sixth root for real numbers.