Final answer:
The inverse function of y = 3√x is y = x^3, which is represented by option A) y = x^3. To find the inverse, one swaps x and y and then resolves for y, resulting in the cube of x as the inverse operation of the cube root. The correct answer is option A.
Step-by-step explanation:
The student is asking for the inverse function of y = 3√x. To find the inverse of a function, we switch the roles of x and y and then solve for y. Starting with y = 3√x, we replace y with x and x with y to get x = 3√y. To isolate y, we raise both sides of the equation to the power of 3, obtaining x^3 = y, which simplifies to y = x^3. Therefore, the inverse function is y = x^3.
Let's briefly discuss some relevant concepts to understand why the answer is correct. An inverse function essentially reverses the original function's effect. If a function f(x) takes an input x and produces an output y, then its inverse function f⁻¹(x) takes y as an input and returns the original input x. This concept is crucial in various fields of mathematics such as algebra and calculus. The inverse of an exponential function like e^x is the natural logarithm ln(x), and for a base-10 exponential function 10^x, the inverse is the base-10 logarithm log₁₀(x).
Considering the options given to the student:
- A) y = x^3 is the correct inverse function.
- B) y = x^1/3 is the original function itself.
- C) y = x^3 + 1 is not the inverse because the additional 1 changes the output.
- D) y = 1/x^3 is the reciprocal of the cube, not the inverse function.