Answer:
The inverse of a function is a function that "undoes" the original function. In other words, if you apply the inverse function to the output of the original function, you will get the original input.
To find the inverse of the function f(x) = 7x^2 + 3 for x >= 0, you can follow these steps:
Replace f(x) with y: y = 7x^2 + 3
Solve for x in terms of y: x = sqrt((y - 3)/7)
Replace x with f^-1(y): f^-1(y) = sqrt((y - 3)/7)
The inverse function, f^-1(y), is defined as follows:
f^-1(y) = sqrt((y - 3)/7) for y >= 3
Note that the inverse function is defined only for y >= 3, because the original function is defined only for x >= 0.
To check that this is the correct inverse function, you can substitute f^-1(y) into the original function and see if you get y:
f(f^-1(y)) = f(sqrt((y - 3)/7))
= 7(sqrt((y - 3)/7))^2 + 3
= 7((y - 3)/7) + 3
= y
Therefore, f(f^-1(y)) = y, which means that f^-1(y) is the inverse function of f(x).