Final answer:
To find the inverse of the function y = 2x² + 7, we must first restrict the domain to ensure the function is one-to-one. After swapping x and y and solving for y, we get two functions representing the inverse: y = ±√((x - 7)/2), which apply to the restricted domain where x ≥ 0.
Step-by-step explanation:
Finding the Inverse of a Function
To find the inverse of the function y = 2x² + 7, we must first recognize that the original function is not one-to-one, which means it does not pass the Horizontal Line Test; each y-value has more than one x-value associated with it due to the squared term. Since the function is not one-to-one, it does not have an inverse function in its entire domain. If we still need to proceed with finding the inverse, we must first restrict the domain of the original function to where it is one-to-one, for example, x ≥ 0 for the function y = 2x² + 7.
If we consider the restricted domain, to find the inverse, we swap x and y and then solve for y. This gives us x = 2y² + 7. To solve for y, we first subtract 7 from both sides to get x - 7 = 2y², and then we divide by 2 to get (x - 7)/2 = y². Finally, we take the square root of both sides, remembering to consider both the positive and negative roots since we are solving for a squared term. This yields y = ±√((x - 7)/2), which are the inverse functions for y = 2x² + 7, restricted to x ≥ 0.